Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )


 Merilyn Owen
 1 years ago
 Views:
Transcription
1 Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O 2 O 3 where O 1, O 2, and O 3 are zero matrices. Thus D i,i = 1 for i r and D i,j = 0 otherwise. Cor 1. Let A be an m n matrix of rank r. Then there exists invertible matrices B M m (F ) and C M n (F ) such that D = BAC, where ( ) Ir O D = 1 O 2 O 3 is the m n matrix in which O 1, O 2, and O 3 are zero matrices. Cor 2. Let A M m n (F ). Then (a) rank(a t ) =rank(a). (b) The rank of any matrix equals the maximum number of its linearly independent rows; that is, the rank of a matrix is the dimension of the subspace generated by its rows. (c) The rows and olumns of any matrix generate subspaces of the same dimension, numerically equal to the rank of the matrix. Cor 3. Every invertible matrix is a product of elementary matrices. Section 2.1 Theorem 2.6. Let V and W be vector spaces over F, and suppose that {v 1, v 2,..., v n } is a basis for V. For w 1, w 2,..., w n in W, there exists exactly one linear transformation T : V W such that T (v i ) = w i for i = 1, 2,..., n. Section 2.4 Theorem Let V and W be finitedimensional vector spaces over F of dimensions n and m, respectively, and let β and γ be ordered bases for V and W, respectively. Then the funtion defined by Φ(T ) = [T ] γ β Φ : L(V, W ) M m n (F ) for T L(V, W ), is an isomorphism. Cor 1. Let V and W be finitedimensional vector spaces over F of dimensions n and m, respectively. Then L(V, W ) is finitedimensional of dimension mn. Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v ) 1
2 2 Theorem Suppose that V is a finitedimensional vector space with the ordered basis β = {x 1, x 2,..., x n }. Let f i (1 i n) be the ith coordinate function with respect to β and let β = {f 1, f 2,..., f)n}. Then β is an ordered basis for V, and, for any f V, we have n f = f(x i )f i. i=1 Theorem Let V and W be finitedimensional vector spaces over F with ordered bases β and γ, respectively. For any linear transformation T : V W, the mapping T t : W V defined by T t (g) = gt for all g W is a linear transformation with the property [T t ] β γ = ([T ]γ β )t. Lemma 1. Let V be a finitedimensional vector space, and let x V. If x = 0 for all f V, then x = 0. Theorem Let V be a finite dimensional vector space, and define ψ : V V by ψ(x) = x. Then ψ is an isomorphism. Cor 1. Let V be a finitedimensional vector space with dual space V. Then every ordered basis for V is the dual basis for some basis for V. Section 4.2 Theorem 4.3. The determinant of an n n matrix is a linear function of each row when the remaining rows are held fixed. Section 5.1: Defn 1. A linear operator T : V V on a finitedimensional vector space V is called diagonalizable if there is an ordered basis β for V such that β [T ] β is a diagonal matrix. A square matrix A is called diagonalizable if L A is diagonalizable. Note 1. If A = B [T ] B where T = L A then to find β such that β [I] B B [T ] B B [I] β = β [T ] β is diagonal is the same thing as finding an invertible S such that S 1 AS is diagonal. Defn 2. Let T be a linear operator on V, a nonzero v V is called an eigenvector of T if λ F, such that T (v) = λv. The scalar λ is called the eigenvalue corresponding to the eigenvector v. Let A M n (F ), v F n, v 0 is called an eigenvector of A if v is an eigenvector of L A. That is, Av = λv. Theorem 5.1. A linear operator T on a finite dimensional vector space V is diagonalizable if and only if there is an ordered basis β for V consisting of eigenvectors of T. Furthermore, if T is diagonalizable, β = {v 1, v 2,..., v n } is an ordered basis of eigenvectors of T, and for D = β [T ] β, D is a diagonal matrix and D j,j is the eval corresponding to v j, 1 j n. Theorem 5.2. A M n (F ). λ is an eval of A if and only if det(a λi) = 0. Defn 3. For matrix A M n (F ), f(t) = det(a ti n ) is the characteristic polynomial of A. Defn 4. Let T be a linear operator on an ndimensional vector space V with ordered basis β. We define the characteristic polynomial f(t) of T to be the characteristic polynomial of A = β [T ] β. That is, f(t) = det(a ti n ).
3 Note 2. Similar matrices have the same characteristic polynomials, since if B = S 1 AS, det(b ti n ) = det(s 1 AS ti n ) = det(s 1 AS ts 1 S) = det(s 1 (A ti n )S) = det(s 1 1 ) det(a ti n ) det(s) = det(s) det(a ti n) det(s) = det(a ti n ) So that characteristic polynomials are similarity invariants. If B 1 and B 2 are 2 bases of V then B1 [T ] B1 is similar to B2 [T ] B2 and so we see that the definition of characteristic polynomial of T, does not depend on the basis used in the representation. So, we may say f(t) = det(t ti n ). Theorem 5.3. Let A M n (F ) (1) The characteristic polynomial of A is a polynomial of degree n with leading coefficient ( 1) n. (2) A has at most n distinct eigenvalues. Theorem 5.4. Let T be a linear operator on a vector space V and let λ be an eval of T. A vector v V is an evctr of T corresponding to λ if and only if v 0 and v N(T λi n ). Section 5.2 Theorem 5.5. Let T be a linear operator on a vector space V and let λ 1, λ 2,..., λ k be distinct evals of T. If v 1, v 2,..., v k are evctrs of T such that for all i [k], λ 1 corresponds to v i, then {v 1, v 2,..., v k } is a linearly independent set. Cor 2. Let T be a linear operator on an ndimensional vector space V. has n distinct evals, then T is diagonlizable. Defn 5. A polynomial f(t) in P (F ) splits over F if there are scalars c, a 1,..., a n such that f(t) = c(t a 1 )(t a 2 ) (t a n ). 3 If T Theorem 5.6. The characteristic polynomial of any diagonalizable linear operator splits. Defn 6. Let λ be an eval of a linear operator (or matrix) with characteristic polynomial f(t). The algebraic multiplicity (or just multiplicity) of λ is the largest positive integer k for which (t λ) k is a factor of f(t). Defn 7. Let T be a linear operator on a vector space V and let λ be an eval of T. Define E λ = {x V T (x = λx} = N(T λi). The set E λ is called the eigenspace of T corresponding to λ. The eigenspace of a matrix A M n (F ) is the espace of L A. Fact 1. E λ is a subspace. Theorem 5.7. Let T be a linear operator on a finite dimensional vector space V, and let λ be an eval of T having multiplicity m. Then 1 dim(e λ ) m. Lemma 2. Let T be a linear operator on a vector space V and let λ 1, λ 2,..., λ k be distinct evals of T. For each i = 1, 2,..., k, let v i E λi. If v 1 + v 2 + v k = 0, then i [k], v i = 0.
4 4 Theorem 5.8. Let T be a linear operator on a vector space V and let λ 1, λ 2,..., λ k be distinct evals of T. For each i = 1, 2,..., k, let S i E λi, be a finite linearly independent set. Then S = S 1 S 2 S k is a linearly independent subset of V. Theorem 5.9. Let T be a linear operator on a finite dimensional vector space V such that the characteristic polynomial of T splits. Let λ 1, λ 2,..., λ k be distinct evals of T. Then (1) T is diagonalizable if and only if the multiplicity of λ i is equal to dim(e λi ), i. (2) If T is diagonalizable and β i is an ordered basis for E λi, for each i, then β = β 1 β 2 β k is an ordered basis for V consisting of eigenvectors of T. Note 3. Test for diagonalization. A linear operator T on a vector space V of dimension n is diagonalizable if and only if both of the following hold. (1) The characteristic polynomial splits. (2) For each λ, eigenvalue of T, the multiplicity of λ equals the dimension of E λ. Notice that E λ = {x (T λ I)(x) = 0} = N(T λi) and n = nullity(t λi) + rank(t λi). So, dim(e λ ) = nullity(t λi) = n rank(t λi). Defn 8. Let W 1, W 2,..., W k be subspaces of a vector space V. The sum is: k W i = {v 1 + v v k : v i W i, i [k]} i=1 Fact 2. The sum is a subspace. Defn 9. Let W 1, W 2,..., W k be subspaces of a vector space V. We call V the direct sum of W 1, W 2,..., W k, written V = W 1 W 2 W k If V is the sum of W 1, W 2,..., W k and j [k], W j i j W i = {0}. Theorem Let W 1, W 2,..., W k be subspaces of a finitedimensional vector space V. Tfae (1) V = W 1 W 2 W k (2) V = k i=1 W i and i, v i W i if v 1 + v 2 + v k = 0 then v i = 0, i (3) Each vector v V can be written uniquely as v = v 1 +v 2 + v k, where i [k], v i W i. (4) If i [k], γ i is an ordered basis for W i then γ 1 γ 2 γ k is an ordered basis for V. (5) For each i [k] there is an ordered basis γ i for W i such that γ 1 γ 2 γ k is an ordered basis for V. Theorem A linear operator T on a finitedimensional vector space V is diagonalizable if and only if V is the direct sum of the eigenspaces of T.
5 Section skipped Section Invariant subspaces and the CayleyHamilton Theorem Defn 1. Let T be a linear operator on a vector space V. A subspace W of V is called a T invariant subspace of V if (W ) W. Defn 2. Let T be a linear operator on a vector space V and let x be a nonzero vector in V. The subspace W = span({x, T (x), T 2 (x),...}) is called the T cyclic subspace of V generated by x. Defn 3. If T is a linear operator on V and W is a T invariant subspace of V, then the restriction T W of T to W is a mapping from W to W and it follows that T W is a linear operator on W. Theorem Let T be a linear operator on a finitedimensional vector space V, and let W be a T invariant subspace of V. Then the caracteristic polynomial of T W divides the characteristic polynomial of T. Theorem Let T be a linear operator on a finitedimensional vector space V, and let W denote the T cyclic subspace of V generated by a nonzero vector v V. Let k = dim(w ). Then (a) {v, T (v), T 2 (v),..., T k 1 (v)} is a basis for W. (b) If a 0 v +a 1 T (v)+ +T k 1 (v) = 0, then the characteristic polynomial of T W is f(t) = ( 1) k (a 0 + a 1 t + + a k 1 t k 1 + t k ). Theorem (CayleyHamilton). Let T be a linear operator on a finitedimensional vector space V, and let f(t) be the characteristic polynomial of T. Then f(t ) = T 0, the zero transformation. Cor 1. (CayleyHamilton Theorem for Matrices). Let A be an n n matrix, and let f(t) be the characteristic polynomial of A. Then f(a) = O, the zero matrix. Theorem Let T be a linear operator on a finitedimensional vector space V, and suppose that V = W 1 W 2 W k, where W i is a T invariant subspace of V for each i (1 i k). Suppose that f i (t) is the characteristic polynomial of T Wi (1 i k). Then f 1 (t) f 2 (t) f k (t) is the characteristic polynomial of T. Defn 4. Let B 1 M m (F ), and let B 2 M n (F ). We define the direct sum of B 1 and B 2, denoted B 1 B 2 as the (m + n) (m + n) matrix A such that (B 1 ) i,j for 1 i, j m A i,j = (B 2 ) (i m),(j m) for m + 1 i, j n + m 0 otherwise If B 1, B 2,..., B k are square matrices with entries from F, then we define the direct sum of B 1, B 2..., B k recursively by B 1 B 2 B k = (B 1 B 2 B k 1 ) B k 5
6 6 Of A = B 1 B 2 B k, then we often write A = B 1 O O O B 2 O... O O B k Theorem Let T be a linear operator on a finitedimensional vector space V, an d let W 1, W 2,..., W k be T invariant subspaces of V such that V = W 1 W 2 W k. For each i, let β i be an ordered basis for W i, and let β = β 1 β 2 β k. Let A = [T ] β and B i = [T Wi ] βi for each i. Then A = B 1 B 2 B k. Section Inner Products and Norms. Defn 5. Let V be a vector space over F. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F, denoted x, y, such that for all x, y, and z in V and all c in F, the following hold: (1) x + z, y = x, y + z, y (2) cx, y = c x, y (3) x, y = y, x, where the bar denotes complex conjugation. (4) x, x > 0 if x 0 Notice that if F = R, (3) is just x, y = y, x. Defn 6. Let A M m n (F ). We define the conjugate transpose or adjoint of A to be the n m matrix A such that (A ) i,j = A i,j for all i, j. Theorem 6.1. Let V be an inner product space. Then for x, y, z V and c F, (1) x, y + z = x, y + x, z (2) x, cy = c x, y (3) x, 0 = 0, x = 0 (4) x, x = 0 x = 0 (5) If x, y = x, z, for all x V, then y = z. Defn 7. Let V be an inner product space. For x V, we define the norm or length of x by x = x, x. Theorem 6.2. Let V be an inner product space over F. Then for all x, y V and c F, the following are true. (1) cx = c x. (2) x = 0 if and only if x = 0. In any case x 0. (3) (CauchySchwarz Inequality) < x, y > x y. (4) (Triangle Inequality) x + y x + y. Defn 8. Let V be an inner product space, x, y V are orthogonal or ( perpendicular ) if x, y = 0. A subset S V is called orthogonal if x, y S, x, y = 0. x V is a unit vector if x = 1 and a subset S V is orthonormal if S is orthogonal and x S, x = 1.
7 Section 6.2 Defn 9. Let V be an inner product space. Then S V is an orthonormal basis of V if it is an ordered basis and orthonormal. Theorem 6.3. Let V be an inner product space and S = {v 1, v 2,..., v k } be an orthogonal subset of V such that i, v i 0. If y Span(S), then k y, v i y = v i v i Cor 1. If S also is orthonormal then k y = y, v i v i i=1 i=1 Cor 2. If S also is orthogonal and all vectors in S are nonzero then S is linearly independent. Theorem 6.4. (GramSchmidt) Let V be an inner product space and S = {w 1, w 2,..., w n } V be a linearly independent set. Define S = {v 1, v 2,..., v n } where v 1 = w 1 and for 2 k n, Then, S v k = w k k 1 i=1 is orthogonal and Span(S ) =Span(S). w k, v i v i. v i Theorem 6.5. Let V be a nonzero finite dimensional inner product space. Then V has an orthonormal basis β. Furthermore, if β = {v 1, v 2,..., v n } and x V, then n x = x, v i v i. i=1 Cor 1. Let V be a finite dimensional inner product space with an orthonormal basis β = {v 1, v 2,..., v n }. Let T be a linear operator on V, and A = [T ] β. Then for any i, j, (A) i,j = T (v j ), v i. Defn 10. Let S be a nonempty subset of an inner product space V. We define S (read S perp ) to be the set of all vectors in V that are orthogonal to every vector in S; that is, S = {x : x, y = 0, y S}. S is called the orthogonal complement of S. Theorem 6.6. Let W be a finite dimensional subspace of an inner product space V, and let y V. Then there exitst unique vectors u W, w W such that y = u+z. Furthermore, if {v 1, v 2..., v k } is an orthonormal basis for W, then k u = y, v i v i. i=1 Cor 1. In the notation of Theorem 6.6, the vector u is the unique vector in W that is closest to y. That is, for any x W, y x y u and we get equality in the previous inequality if and only if x = u. 7
8 8 Theorem 6.7. Suppose that S = {v 1, v 2,..., v k } is an orthonormal set in an ndimensional inner product space V. Then S can be extended to an orthonormal basis {v 1,..., v k, v k+1,..., v n } If W =Span(S) then S 1 = {v k+1, v k+2,..., v b } is an orthonormal basis for W. If W is any subspace of V, then dim(v ) = dim(w )+ dim(w ). Section 6.3: Theorem 6.8. Let V be a finite dimensional inner product space over F, and let g : V F be a linear functional. Then, there exists a unique vector y V such that g(x) = x, y, x V. Theorem 6.9. Let V be a finite dimensional inner product space and let T be a linear operator on V. There exists a unique operator T : V V such that T (x), y = x, T (y), x, y V. Furthermore, T is linear. Defn 11. T is called the adjoint of the linear operator T and is defined to be the unique operator on V satisfying T (x), y = x, T (y), x, y V. Fact 3. x, T (y) = T (x), y, x, y V. Theorem Let V be a finite dimensional inner product space and let β be an orthonormal basis for V. If T is a linear operator on V, then [T ] β = [T ] β Cor 2. Let A be an n n matrix, then L A = (L A ). Theorem Let V be an inner product space, T, U linear operators on V. Then (1) (T + U) = T + U. (2) (ct ) = ct, c F. (3) (T U) = U T (composition) (4) (T ) = T (5) I = I. Cor 1. Let A and B be n n matrices. (1) (A + B) = A + B. (2) (ca) = ca, c F. (3) (AB) = B A (4) (A ) = A (5) I = I. Then Example 2. (Exercise 5b) Let A and B be m n matrices and C an n p matrix. Then (1) (A + B) = A + B. (2) (ca) = ca, c F. (3) (AC) = C A (4) (A ) = A (5) I = I. For x, y F n, let x, y n denote the standard inner product of x and y in F n. Recall that if x and y are regarded as column vectors, then x, y n = y x.
9 Lemma 3. Let A M m n (F ), x F n, and y F m. Then Ax, y m = x, A y n. Lemma 4. Let A M m n (F ), x F n. Then rank(a A) =rank(a). Cor 1. If A is an m n matrix such that rank(a) = n, then A A is invertible. Theorem Let A M m n (F ) and y F m. Then there exists x 0 F n such that (A A)x 0 = A y and Ax 0 y Ax y for all x F n. Furthermore, if rank(a) = n, then x 0 = (A A) 1 A y. Section 6.4: Lemma 5. Let T be a linear operator on a finitedimensional inner product space V. If T has an eigenvector, then so does T. Theorem (Schur). Let T be a linear operator on a finitedimensional inner product space V. Suppose that the characteristic polynomial of T splits. Then there exists an orthonormal basis β for V such that the matrix [T ] β is upper triangular. Defn 12. Let V be an inner product space, and let T be a linear operator on V. We say that T is normal if T T = T T. An n n real or complex matrix A is normal if AA = A A. Theorem Let V be an inner product space, and let T be a normal operator on V. Then the following statements are true. (1) T (x) = T (x) for all x V. (2) T ci is normal for every c F (3) If x is an eigenvector of T, then x is also an eigenvector of T. In fact, if T (x) = λx, then T (x) = λx. (4) If λ 1 and λ 2 are distinct eigenvalues of T with corresponding eigenvectors x 1 and x 2, then x 1 and x 2 are orthogonal. Defn 13. Let T be a linear operator on an inner product space V. We say that T is selfadjoint ( Hermitian) if T = T. An n n real or complex matrix A is selfadjoint (Hermitian) if A = A. Theorem Let T be a linear operator on a finitedimensional complex inner product space V. Then T is normal if and only if there exists an orthonormal basis for V consisting of eigenvectors of T. Lemma 6. Let T be a selfadjoint operator on a finitedimensional inner product space V. Then (1) Every eigenvalue of T is real. (2) Suppose that V is a real inner product space. Then the characteristic polynomial of T splits. Theorem Let T be a linear operator on a finitedimensional real inner product space V. Then T is selfadjoint if and only if there exists an orthonormal basis β for V consisting of eigenvectors of T. 9
10 10 Section 6.5: Defn 14. Let T be a linear operator on a finitedimensional inner product space V (over F ). If T (x) = x for all x V, we call T a unitary operator if F = C and an orthogonal operator if F = R. (Comment on relationship to onetoone, onto, and invertible.) Theorem Let T be a linear operator on a finitedimensional inner product space V. Then the following statements are equivalent. (1) T T = T T = I. (2) T (x), T (y) = x, y, x, y V. (3) If β is an orthonormal basis for V, then T (β) is an orthonormal basis for V. (4) There exists an orthonormal basis β for V such that T (β) is an orthonormal basis for V. (5) T (x) = x, x V. Lemma 7. Let U be a selfadjoint operator on a finitedimensional inner product space V. If x, U(x) = 0 for all x V, then U = T 0. Cor 1. Let T be a linear operator on a finitedimensional real inner product space V. Then V has an orthonormal basis of eigenvectors of T with corresponding eigenvalues of absolute value 1 if and only if T is both selfadjoint and orthogonal. Cor 2. Let T be a linear operator on a finitedimensional complex inner product space V. Then V has an orthonormal basis of eigenvectors of T with corresponding eigenvalues of absolute value 1 if and only if T is unitary. Defn 15. A square matrix A is called an orthogonal matrix if A t A = AA t = I and unitary if A A = AA = I. Theorem Let A be a complex n n matrix. Then A is normal if and only if A is unitarily equivalent to a diagonal matrix. Theorem Let A be a real n n matrix. Then A is symmetric if and only if A is orthogonally equivalent to a real diagonal matrix. Theorem Let A M n (F ) be a matrix whose characteristic polynomial splits over F. (1) If F = C, then A is unitarily equivalent to a complex upper triangular matrix. (2) If F = R, then A is orthogonally equivalent to a real upper triangular matrix. Section 6.6 Defn 16. If V = W 1 W 2, then a linear operator T on V is the projection on W 1 along W 2 if whenever x = x 1 + x 2, with x 1 W 1 and x 2 W 2, we have T (x) = x 1. In this case, R(T ) = W 1 = {x V : T (x) = x} and N(T ) = W 2.
11 We refer to T as the projection. Let V be an inner product space, and let T : V V be a projection. We say that T is an orthogonal projection if R(T ) = N(T ) and N(T ) = R(T ). Theorem Let V be an inner product space, and let T be a linear operator on V. Then T is an orthogonal projection if and only if T has an adjoint T and T 2 = T = T. (Note compare to Theorem 6.9 where V is finitedimensional) Theorem (The Spectral Theorem). Suppose that T is a linear operator on a finitedimensional inner product space V over G with the distinct eigenvalues λ 1, λ 2,..., λ k. Assume that T is normal if F = C and that T is selfadjoint if F = R. For each i(1 i k), let W i be the eigenspace of T corresponding to the eigenvalue λ i, and let T i be the orthogonal projection of V on W i. Then the following statements are true. (1) V = W 1 W 2 W k. (2) If W i denotes the direct sum of the subspaces W j for j i, then Wi = W i. (3) T i T j = δ i,j T i for 1 i, j k. (4) I = T 1 + t T k. (5) T = λ 1 T 1 + λ 2 T λ k T k. Defn 17. The set {λ 1, λ 2,..., λ k } of eigenvalues of T is called the spectrum of T, the sum I = T 1 + T T k is called the resolution of the identity operator induced by T, and the sum T = λ 1 T 1 + λ 2 T λ k T k is called the spectral decomposision of T. Cor 1. If F = C, then T is normal if and only if T = g(t ) for some polynomial g. Cor 2. If F = C, then T is unitary if and only if T is normal and λ = 1 for every eigenvalue λ of T. Cor 3. If F = C and T is normal, then T is selfadjoint if and only if every eigenvalue of T is real. Cor 4. Let T be as in the spectral theorem with spectral decomposition T = λ 1 T 1 + λ 2 T λ k T k. Then each T j is a polynomial in T. Section 6.7 Theorem (Singular Value Theorem for Linear Transformations). Let V and W be finitedimensional inner product spaces, and let T : V W be a linear transformation of rank r. Then there exist orthonormal bases {v 1, v 2,..., v n } for V and {u 1, u 2,..., u m } for W and positive scalars σ 1 σ 2 σ r such that { σi u T (v i ) = i if 1 i r 0 if i > r Conversely, suppose that the preceding conditions are satisfied. Then for 1 i n, v i is an eigenvector of T T with corresponding eigenvalue σi 2 if 1 i r and 0 if i > r. Therefore the scalars σ 1, σ 2,..., σ r are uniquely determined by T. 11
12 12 Defn 18. The unique scalars σ 1, σ 2,..., σ r in Theorem 6.26 are called the singular values of T. If r is less than both m and n, then the term singular value is extended to include σ r+1 = = σ k = 0, where k is the minimum of m and n. Defn 19. Let A be an m n matrix. We define the singular values of A to be the singular values of the linear transformation L A. Theorem (Singular Value Decomposition Theorem for Matrices). Let A be an m n matrix of rank r with the positive singular values σ 1 σ 2 σ r, and let Σ be the m n matrix defined by { σi if i = j r Σ i,j = 0 otherwise Then there exists an m n unitary matrix U and an n n unitary matrix V such that A = UΣV. Defn 20. Let A be an m n matrix of rank r with positive singular values σ 1 σ 2 σ r. A factorization A = UΣV where U and V are unitary matrices and Σ is the m n matrix defined as in Theorem 6.27 is called a singular value decomposition of A.
13 Section 7.1: Defn 1. A square matrix A i is called a Jordan block corresponding to λ if it is of the form: λ λ A i = λ λ Defn 2. A square matrix [T ] β is called a Jordan canonical form of T if it has the following form, where each A i is a Jordan block and each O is the zero matrix. A 1 O O O O O A 2 O O O [T ] β =..... O O O A k 1 O O O O O A k The basis β used to form the Jordan canonical form of T is called the Jordan canonical basis for T. Defn 3. Let T be a linear operator on a vector space V, and let λ be a scalar. A nonzero vector x in V is called a generalized eigenvector of T corresponding to λ if (T λi) p (x) = 0 for some positive integer p. Defn 4. Let T be a linear operator on a vector space V, and let λ be an eigenvalue of T. The generalized eigenspace of T corresponding to λ, denoted K λ, is the subset of V defined by K λ = {x V : p Z, (T λi) p (x) = 0} Theorem 7.1. Let T be a linear operator on a vector space V, and let λ be an eigenvalue of T. Then (1) K λ is a T invariant subspace of V containing E λ. (2) For any scalar µ λ, the restriction of T µi to K λ is onetoone. Theorem 7.2. Let T be a linear operator on a finitedimensional vector space V such that the characteristic polynomial of T splits. Suppose that λ is an eigenvalue of T with multiplicity of m. Then (1) dim(k λ ) m. (2) K λ = N((T λi) m ). Theorem 7.3. Let T be a linear operator on a finitedimensional vector space V such that the characteristic polynomial of T splits, and let λ 1, λ 2,..., λ k be the distinct eigenvalues of T. Then, for every x V, there exist vectors v i K λi, 1 i k, such that x = v 1 + v v k. 13
14 14 Theorem 7.4. Let T be a linear operator on a finitedimensional vector space V such that the characteristic polynomial of T splits, and let λ 1, λ 2,..., λ k be the distinct eigenvalues of T corresponding to the multiplicities m 1, m 2,..., m k. For 1 i k, let β i be an ordered basis for K λi. Then the following statements are true. (a) β i β j = for i j. (b) β = β 1 β 2 β k is an ordered basis for V. (c) dim(k λi ) = m i, for all i. Cor 5. Let T be a linear operator on a finitedimensional vector space V such that the characteristic polynomial of T splits. Then T is diagonalizable if and only if E λ = K λ for every eigenvalue λ of T. Defn 5. Let T be a linear operator on a vector space V, and let x be a generalized eigenvector of T corresponding to the eigenvalue λ. Suppose that p is the smallest positive integer for which (T λi) p (x) = 0. Then the ordered set {(T λi) p 1 (x), (T λi) p 2 (x),..., (T λi)(x)} is called a cycle of generalized eigenvectors of T corresponding to λ. The vectors (T λi) p 1 (x) and x are called the initial vector and the end vector of the cycle, respectively. We say that the length of the cycle is p. Theorem 7.5. Let T be a linear operator on a finitedimensional vector space V whose characteristic polynomial splits, and suppose that β is a basis for V such that β is a disjoint union of cycles of generalized eigenvectors of T. Then the following statements are true. (a) For each cycle γ of generalized eigenvectors contained in β, W =span(γ) is T invariant, and [T W ] γ is a Jordan block. (b) β is a Jordan canonical basis for V. Theorem 7.6. Let T be a linear operator on a vector space V, and let λ be an eigenvalue of T. Suppose that γ 1, γ 2,..., γ q are cycles of generalized eigenvectors of T corresponding to λ such that the initial vectors of the γ i s are distinct and form a linearly independent set. then the γ i s are disjoint, and their union γ = q i=1 γ i is linearly independent. Cor 1. Every cycle of generalized eigenvectors of a linear operator is linearly independent. Theorem 7.7. Let T be a linear operator on a finitedimensional vector space V, and let λ be an eigenvalue of T. Then K λ has an ordered basis consisting of a union of disjoint cycles of generalized eigenvectors corresponding to λ. Cor 1. Let T be a linear operator on a finitedimensional vector space V whose characteristic polynomial splits. Then T has a Jordan canonical form. Defn 6. Let A M n (F ) be such that the characteristic polynomial of A (and hence of L A ) splits. Then the Jordan canonical form of A is defined to be the Jordan canonical form of the linear operator L A on F n.
15 Cor 2. Let A be an n n matrix whose characteristic polynomial splits. Then A has a Jordan canonical form J, and A is similar to J. Theorem 7.8. Let T be a linear operator on a finitedimensional vector space V whose characteristic polynomial splits. Then V is the direct sum of the generalized eigenspaces of T. (Compare to Theorem 5.11) 15
16 16 Section 7.2 For the purposes of this section, we fix a linear operator T on an ndimensional vector space V such that the characteristic polynomial of T splits. Let λ 1, λ 2,..., λ k be the distinct eigenvalues of T. By Theorem 7.7, each generalized eigenspace K λi contains an ordered basis β i consisting of a union of disjoint cycles of generalized eigenvectors corresponding to λ i. So by Theorems 7.4(b) and 7.5, the union β = β i is a Jordan canonical basis for T. For each i, let T i be the restriction of T to K λi, and let A i = [T i ] βi. Then A i is the Jordan canonical form of T i, and A 1 O O O A 2 O J = [T ] β =... O O A k is the Jordan canonical form of T. In this matrix, each O is a zero matrix of appropriate size. Defn 1. We adopt the following convention: The basis β i for K λi will henceforth be ordered in such a way that the cycles appear in order of decreasing length. That is, if β i is a disjoint union of cycles γ 1, γ 2,..., γ ni and if the length of the cycle γ j is p j, we index the cycles so that p 1 p 2 p ni. This ordering of the cycles limits the possible orderings of vectors in β i, which in turn determines the matrix A i. It is in this sense that A i is unique. It then follows that the Jordan canonical form for T is unique up to an ordering of the eigenvalues of T. Defn 2. Let T i be the restriction of T to K λi. A dot diagram of T i will aid in the visualization of the matrix A i. Suppose that β i is a disjoint union of cycles of generalized eigenvectors γ 1, γ 2,..., γ ni with lengths p 1 p 2 p ni, respectively. The dot diagram contains one dot for each vector in β i, and the dots are configured according to the following rules. (1) The array consists of n i columns (one column for each cycle.) (2) Counting from left to right, the j th column consists of the p j dots that correspond to the vectors of γ j starting with the initial vector at the top and continuing down to the end vector. Denote the end vectors of the cycles by v 1, v 2,..., v ni. In the following dot diagram of T i, each dot is labeled with the name of the vector in β i to which it corresponds. (T λ i I) p1 1 (v 1 ) (T λ i I) p2 1 (v 2 ) (T λ i I) p n 1 i (v ni ) (T λ i I) p1 2 (v 1 ) (T λ i I) p2 2 (v 2 ) (T λ i I) pn i 2 (v ni ). (T λ i I)(v 1 ) v 1. (T λ i I)(v 2 ) v 2. (T λ i I)(v ni ) v ni Theorem 7.9. We fix a basis β i of K λi so that β i is a disjoint union of n i cycles of generalized eigenvectors with lengths p 1 p 2 p ni. For any positive integer r, the vectors in β i
17 that are associated with the dots in the first r rows of the dot diagram of T i constitute a basis for N((T λ i I) r ). Hence the number of dots in the first r rows of the dot diagram equals nullity((t λ i I) r ). Cor 1. The dimension of E λi is n i. Hence in a Jordan canonical form of T, the number of Jordan blocks corresponding to λ i equals the dimension of E λi. Theorem Let r i denote the number of dots in the j th row of the dot diagram of T i, the restriction of T to K λi. Then the following statements are true. (a) r 1 = dim(v ) rank(t λ i I). (b) r j = rank((t λ i I) j 1 ) rank((t λ i I) j ), if j > 1. Cor 1. For any eigenvalue λ i of T, the dot diagram of T i is unique. Thus, subject to the convention that the cycles of generalized eigenvectors for the bases of each generalized eigenspace are listed in order of decreasing length, the Jordan canonical form of a linear operator or a matrix is unique up to the ordering of the eigenvalues. Theorem Let A and B be n n matrices, each having Jordan canonical forms computed according to the conventions of this section. Then A and B are similar if and only if they have (up to an ordering of their eigenvalues) the same Jordan canonical form. Section 7.3 Defn 1. Let T be a linear operator on a finitedimensional vector space. A polynomial p(t) is called a minimal polynomial of T if p(t) is a monic polynomial of least positive degree for which p(t ) = T 0. Theorem Let p(t) be a minimal polynomial of linear operator T on a finitedimens vector space V. (a) For any polynomial g(t), if g(t ) = T 0, then p(t) divides g(t). p(t) divides the characteristic polynomial of T. (b) The minimal polynomial of T is unique. 17 In particular, Defn 2. Let A M n (F ). The minimal polynomial p(t) of A is the monic polynomial of least positive degree for which p(a) = O. Theorem Let T be a linear operator on a finitedimensional vector space V, and let β be an ordered basis for V. Then the minimal polynomial of T is the same as the minimal polynomial of [T ] β. Cor 1. For any A M n (F ), the minimal polynomial of A is the same as the minimal polynomial of L A. Theorem Let T be a linear operator on a finitedimensional vector space V, and let p(t) be the minimal polynomial of T. A scalar λ is an eigenvalue of T if and only if p(λ) = 0. Hence the characteristic polynomial and the minimal polynomial of T have the same zeros. Cor 1. Let T be a linear operator on a finitedimensional vector space V with minimal polynomial p(t) and characteristic polynomial f(t). Suppose that f(t) factors as f(t) = (λ 1 t) n 1 (λ 2 t) n2 (λ k t) n k,
18 18 where λ 1, λ 2,..., λ k are the distinct eigenvalues of T. Then there exist integers m 1, m 2,..., m k such that 1 m i n i, for all i and p(t) = (t λ 1 ) m 1 (t λ 2 ) m2 (t λ k ) m k. Theorem Let T be a linear operator on a finitedimensional vector space V such that V is a T cyclic subspace of itself. Then the characteristic polynomial f(t) and the minimal polynomial p(t) have the same degree, and hence f(t) = ( 1) n p(t). Theorem Let T be a linear operator on a finitedimensional vector space V. Then G is diangonalizable if and only if the minimal polynomial of T is of the form p(t) = (t λ 1 )(t λ 2 ) (t λ k ) where λ 1, λ 2,..., λ k are the distinct eigenvalues of T.
MATH 235. Final ANSWERS May 5, 2015
MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationLinear Algebra 1. M.T.Nair Department of Mathematics, IIT Madras. and in that case x is called an eigenvector of T corresponding to the eigenvalue λ.
Linear Algebra 1 M.T.Nair Department of Mathematics, IIT Madras 1 Eigenvalues and Eigenvectors 1.1 Definition and Examples Definition 1.1. Let V be a vector space (over a field F) and T : V V be a linear
More informationMATRICES ARE SIMILAR TO TRIANGULAR MATRICES
MATRICES ARE SIMILAR TO TRIANGULAR MATRICES 1 Complex matrices Recall that the complex numbers are given by a + ib where a and b are real and i is the imaginary unity, ie, i 2 = 1 In what we describe below,
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationThen x 1,..., x n is a basis as desired. Indeed, it suffices to verify that it spans V, since n = dim(v ). We may write any v V as r
Practice final solutions. I did not include definitions which you can find in Axler or in the course notes. These solutions are on the terse side, but would be acceptable in the final. However, if you
More informationChapter 6 Inner product spaces
Chapter 6 Inner product spaces 6.1 Inner products and norms Definition 1 Let V be a vector space over F. An inner product on V is a function, : V V F such that the following conditions hold. x+z,y = x,y
More informationLinear Algebra Highlights
Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to
More informationBASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x
BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,
More informationContents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2
Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Page of 7 Linear Algebra Practice Problems These problems cover Chapters 4, 5, 6, and 7 of Elementary Linear Algebra, 6th ed, by Ron Larson and David Falvo (ISBN3 = 97868783762,
More informationDefinitions for Quizzes
Definitions for Quizzes Italicized text (or something close to it) will be given to you. Plain text is (an example of) what you should write as a definition. [Bracketed text will not be given, nor does
More informationMATHEMATICS 217 NOTES
MATHEMATICS 27 NOTES PART I THE JORDAN CANONICAL FORM The characteristic polynomial of an n n matrix A is the polynomial χ A (λ) = det(λi A), a monic polynomial of degree n; a monic polynomial in the variable
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationLinear Algebra Lecture NotesII
Linear Algebra Lecture NotesII Vikas Bist Department of Mathematics Panjab University, Chandigarh64 email: bistvikas@gmail.com Last revised on March 5, 8 This text is based on the lectures delivered
More information235 Final exam review questions
5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342  Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationLinear Algebra. Workbook
Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx
More informationMath 24 Spring 2012 Sample Homework Solutions Week 8
Math 4 Spring Sample Homework Solutions Week 8 Section 5. (.) Test A M (R) for diagonalizability, and if possible find an invertible matrix Q and a diagonal matrix D such that Q AQ = D. ( ) 4 (c) A =.
More informationFinal A. Problem Points Score Total 100. Math115A Nadja Hempel 03/23/2017
Final A Math115A Nadja Hempel 03/23/2017 nadja@math.ucla.edu Name: UID: Problem Points Score 1 10 2 20 3 5 4 5 5 9 6 5 7 7 8 13 9 16 10 10 Total 100 1 2 Exercise 1. (10pt) Let T : V V be a linear transformation.
More informationMATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS
MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS There will be eight problems on the final. The following are sample problems. Problem 1. Let F be the vector space of all real valued functions on
More informationMath 108b: Notes on the Spectral Theorem
Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner product space V has an adjoint. (T is defined as the unique linear operator
More informationMATH 115A: SAMPLE FINAL SOLUTIONS
MATH A: SAMPLE FINAL SOLUTIONS JOE HUGHES. Let V be the set of all functions f : R R such that f( x) = f(x) for all x R. Show that V is a vector space over R under the usual addition and scalar multiplication
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW JC Stuff you should know for the exam. 1. Basics on vector spaces (1) F n is the set of all ntuples (a 1,... a n ) with a i F. It forms a VS with the operations of + and scalar multiplication
More informationEigenvalues and Eigenvectors A =
Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector
More informationLast name: First name: Signature: Student number:
MAT 2141 The final exam Instructor: K. Zaynullin Last name: First name: Signature: Student number: Do not detach the pages of this examination. You may use the back of the pages as scrap paper for calculations,
More informationLinear Algebra in Actuarial Science: Slides to the lecture
Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a ToolBox Linear Equation Systems Discretization of differential equations: solving linear equations
More informationMath 489AB Exercises for Chapter 2 Fall Section 2.3
Math 489AB Exercises for Chapter 2 Fall 2008 Section 2.3 2.3.3. Let A M n (R). Then the eigenvalues of A are the roots of the characteristic polynomial p A (t). Since A is real, p A (t) is a polynomial
More informationa 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12
24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2
More informationMath 554 Qualifying Exam. You may use any theorems from the textbook. Any other claims must be proved in details.
Math 554 Qualifying Exam January, 2019 You may use any theorems from the textbook. Any other claims must be proved in details. 1. Let F be a field and m and n be positive integers. Prove the following.
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationGQE ALGEBRA PROBLEMS
GQE ALGEBRA PROBLEMS JAKOB STREIPEL Contents. Eigenthings 2. Norms, Inner Products, Orthogonality, and Such 6 3. Determinants, Inverses, and Linear (In)dependence 4. (Invariant) Subspaces 3 Throughout
More informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationLINEAR ALGEBRA BOOT CAMP WEEK 2: LINEAR OPERATORS
LINEAR ALGEBRA BOOT CAMP WEEK 2: LINEAR OPERATORS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationChapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors
Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More information6 Inner Product Spaces
Lectures 16,17,18 6 Inner Product Spaces 6.1 Basic Definition Parallelogram law, the ability to measure angle between two vectors and in particular, the concept of perpendicularity make the euclidean space
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationDIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix
DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationProblem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show
MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology  Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its ApplicationsLab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
More informationEE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2
EE/ACM 150  Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko
More informationj=1 x j p, if 1 p <, x i ξ : x i < ξ} 0 as p.
LINEAR ALGEBRA Fall 203 The final exam Almost all of the problems solved Exercise Let (V, ) be a normed vector space. Prove x y x y for all x, y V. Everybody knows how to do this! Exercise 2 If V is a
More informationLINEAR ALGEBRA MICHAEL PENKAVA
LINEAR ALGEBRA MICHAEL PENKAVA 1. Linear Maps Definition 1.1. If V and W are vector spaces over the same field K, then a map λ : V W is called a linear map if it satisfies the two conditions below: (1)
More informationNATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS SEMESTER 2 EXAMINATION, AY 2010/2011. Linear Algebra II. May 2011 Time allowed :
NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS SEMESTER 2 EXAMINATION, AY 2010/2011 Linear Algebra II May 2011 Time allowed : 2 hours INSTRUCTIONS TO CANDIDATES 1. This examination paper contains
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More informationLinear algebra and applications to graphs Part 1
Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces
More information(v, w) = arccos( < v, w >
MA322 Sathaye Notes on Inner Products Notes on Chapter 6 Inner product. Given a real vector space V, an inner product is defined to be a bilinear map F : V V R such that the following holds: For all v
More informationELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices
ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data TwoDimensional Plots Programming in
More informationLinear Algebra Final Exam Review
Linear Algebra Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
More informationDSGA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DSGA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More information5. Diagonalization. plan given T : V V Does there exist a basis β of V such that [T] β is diagonal if so, how can it be found
5. Diagonalization plan given T : V V Does there exist a basis β of V such that [T] β is diagonal if so, how can it be found eigenvalues EV, eigenvectors, eigenspaces 5.. Eigenvalues and eigenvectors.
More informationMATH 31  ADDITIONAL PRACTICE PROBLEMS FOR FINAL
MATH 3  ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left
More informationMath 110 Linear Algebra Midterm 2 Review October 28, 2017
Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections
More informationLinear Algebra Review
Chapter 1 Linear Algebra Review It is assumed that you have had a course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc. I will review some of these terms here, but quite
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More information1 Last time: leastsquares problems
MATH Linear algebra (Fall 07) Lecture Last time: leastsquares problems Definition. If A is an m n matrix and b R m, then a leastsquares solution to the linear system Ax = b is a vector x R n such that
More informationLinGloss. A glossary of linear algebra
LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasitriangular A matrix A is quasitriangular iff it is a triangular matrix except its diagonal
More informationLINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS
LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts (in
More information1 9/5 Matrices, vectors, and their applications
1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric
More informationEcon Slides from Lecture 7
Econ 205 Sobel Econ 205  Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for
More informationSYLLABUS. 1 Linear maps and matrices
Dr. K. Bellová Mathematics 2 (10PHYBIPMA2) SYLLABUS 1 Linear maps and matrices Operations with linear maps. Prop 1.1.1: 1) sum, scalar multiple, composition of linear maps are linear maps; 2) L(U, V
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linearalgebrasummarysheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationFall TMA4145 Linear Methods. Exercise set Given the matrix 1 2
Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an
More informationKnowledge Discovery and Data Mining 1 (VO) ( )
Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory
More informationUNIT 6: The singular value decomposition.
UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 20112012 A square matrix is symmetric if A T
More informationLecture 2: Linear operators
Lecture 2: Linear operators Rajat Mittal IIT Kanpur The mathematical formulation of Quantum computing requires vector spaces and linear operators So, we need to be comfortable with linear algebra to study
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More informationMath 413/513 Chapter 6 (from Friedberg, Insel, & Spence)
Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence) David Glickenstein December 7, 2015 1 Inner product spaces In this chapter, we will only consider the elds R and C. De nition 1 Let V be a vector
More information33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM
33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM (UPDATED MARCH 17, 2018) The final exam will be cumulative, with a bit more weight on more recent material. This outline covers the what we ve done since the
More informationChapter 4 Euclid Space
Chapter 4 Euclid Space Inner Product Spaces Definition.. Let V be a real vector space over IR. A real inner product on V is a real valued function on V V, denoted by (, ), which satisfies () (x, y) = (y,
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationLec 2: Mathematical Economics
Lec 2: Mathematical Economics to Spectral Theory Sugata Bag Delhi School of Economics 24th August 2012 [SB] (Delhi School of Economics) Introductory Math Econ 24th August 2012 1 / 17 Definition: Eigen
More informationOnline Exercises for Linear Algebra XM511
This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2
More informationLinear Algebra. Session 12
Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)
More information(f + g)(s) = f(s) + g(s) for f, g V, s S (cf)(s) = cf(s) for c F, f V, s S
1 Vector spaces 1.1 Definition (Vector space) Let V be a set with a binary operation +, F a field, and (c, v) cv be a mapping from F V into V. Then V is called a vector space over F (or a linear space
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show
More informationMATH 221, Spring Homework 10 Solutions
MATH 22, Spring 28  Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the
More informationPreliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012
Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.
More informationLINEAR ALGEBRA 1, 2012I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationMATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
More information1. In this problem, if the statement is always true, circle T; otherwise, circle F.
Math 1553, Extra Practice for Midterm 3 (sections 4565) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation
More informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More information