Linear algebra 2. Yoav Zemel. March 1, 2012


 Arthur Williams
 2 years ago
 Views:
Transcription
1 Linear algebra 2 Yoav Zemel March 1, 2012 These notes were written by Yoav Zemel. The lecturer, Shmuel Berger, should not be held responsible for any mistake. Any comments are welcome at Linear transformations in an inner product space Let V be an inner product space and T : V V a linear transformation. We wish to define its conjugate transformation T : V V. The transformation (α, β) T (α), β is the bilinear pattern defined by T. Lemma 3.1 Let α, β, γ, δ in V, and a, b in F, then (1) T (α + γ), β = T (α), β + T (γ), β. (2) T (α), β + δ = T (α), β + T (α), δ. (3) T (aα), β = a T (α), β. (4) T (α), bβ = b T (α), β. Proof. Trivial by linearity of T and the definition of the inner product. Lemma 3.2 Let β, γ in V. (1) If for any α V α, β = α, γ, or for any α V β, α = γ, α then γ = β. (2) Let T and S be linear transformations V V. If for any α, β V T (α), β = S(α), β or for any α α, T (β) = α, S(β) then T = S. (3) If for any α, β V we have T (α), β = 0 or for any α, β V we have α, T (β) = 0 then T = 0. Proof. (1) If for any α α, β γ = 0 then β γ is orthogonal to all vectors, and therefore equals 0. The other case is analogous. 1
2 (2) If T (α) S(α), β = 0 for any β, then T (α) S(α) = 0. This holds for any α, hence T = S. the other case is analogous. (3) We have T (α), β = 0 = S(α), β, where S is the zero transformation. By (2) T = 0, and the other case is analogous. So T determines a bilinear pattern T (α), β. By this lemma, this linear pattern determines T. If T (α), α = 0 for any α, it does not necessarily mean that T = 0, if F = R e.g. when T is rotation by 90 degrees. However, if F = C then it is true. Proof. For any α, β V we have 0 = T (α + β), α + β = T (α), α + T (α), β + T (β), α + T (β), β = T (α), β + T (β), α. Replacing β by iβ and using i = i we get i T (β), α i T (α), β = 0. On the other hand, we also have i T (β), α + i T (α), β = 0. From this it follows that T (β), α = 0 for any β and any α, and thus T = 0. Let β V and consider the linear functional ϕ β : V R defined by ϕ β (α) = α, β. Note that ϕ β = β, α is linear only over R, not over C. Theorem 3.3 For any linear functional ϕ : V F there exists a unique β for which ϕ = ϕ β. Note : this is true only when dimv <, e.g. V =continuous functions on [1,1] and the functional ϕ(f) = f(0) does not come from any inner product. Proof. Suppose that ϕ β = ϕ γ then for any α, α, β = α, γ and by a previous lemma γ = β. This establishes uniqueness. For the existence, we consider all the linear functionals ϕ β for β V. This is a subspace of the dual space V. As dimv = dimv, we need to show that n = dimv = dim{ϕ β β V } and this will establish V = {ϕ β β V }. Let S : V V be defined by S(β) = ϕ β. S is linear over R, and almost linear over C : S(β 1 +β 2 )(α) = ϕ β1 +β 2 (α) = α, β 1 + β 2 = α, β 1 + α, β 2 = ϕ β1 (α)+ϕ β2 (α). Thus S(β 1 + β 2 ) = S(β 1 ) + S(β 2 ). But S(cβ)(α) = α, cβ = c α, β = cs(β)(α) cs(β)(α), 2
3 so S is not necessarily linear. S is injective because if ϕ β = ϕ γ then β = γ. We know that if dimv = n and T : V W is injective then its image T (V ) is a subspace of W with dimension n. The same proof will work here for S, even though it is not linear but almost linear. It follows that Im(S) = V and the existence proof is complete. Theorem 3.4 conjugate transformation Let V be an inner product space of dimension n <. For any linear transformation T : V V there exists a unique linear transformation T : V V, such that α, β T (α), β = α, T (β). T is the conjugate transformation of T. Proof. (uniqueness) If α, β α, T (β) = α, S (β), then T = S by a previous lemma. (existence) Let β V and consider ϕ β (α) = T α, β. By theorem 3 there exists γ V such that for any α, ϕ β (α) = α, γ. Define T (β) = γ, which is well defined by uniqueness of γ. Then α we have T α, β = α, T (β). Do the same procedure for any β to define T over whole V. All that remains is to show that T is linear. Let α, β and γ in V, then α, T (β + γ) = T α, β + γ = T α, β + T α, γ = α, T (β) + α, T (γ) = α, T (β) + T (γ). This holds for any α, and by lemma 2A, T (β + γ) = T (β) + T (γ). For a scalar s F, α, T (sβ) = T α, sβ = s T (α), β = s α, T (β) = α, st (β). This, again, holds for any α and therefore T (sβ) = st (β). T is linear, and the proof is complete. Theorem 3.5 Let T : V V and let T be its conjugate. Let {e 1,..., e n } be an orthonormal basis of V under which A is the matrix that represents T. Then the matrix that represents T is the matrix A defined by A ij = A ji i, j. A is the conjugate matrix of A. Proof. The j th column of A is the coordinate vector of T (e j ) by the given basis. The i th element of it is A ij. We also know that the i th element equals T (e j ), e i. Thus A ij = T (e j ), e i = e i, T (e j ) = T (e i ), e j = A ji. 3
4 Consider C 2 with the standard inner product, and T (z 1, z 2 ) = (z 1 + iz 2, 2z 2 + (2 3i)z 1 ). Then T (1, 0) = (1, 2 3i) = 1(1, 0)+(2 3i)(0, 1) and T (0, 1) = (i, 2) = i(1, 0)+2(0, 1), so the matrix of T is therefore ( ) 1 i A = 2 3i 2 and A = ( ) i. i 2 This gives T (z 1, z 2 ) = (z 1 + (2 + 3i)z 2, 2z 2 iz 1 ). 3.1 Properties of the conjugate transformation Consider the map : hom(v, V ) hom(v, V ). Lemma 3.6 (1) (T + S) = T + S. (2) (at ) = at. (3) (T S) = S T. (4) (T ) = T. (5) The * transformation is bijective. Proof. Rather immediate : For (1), observe that α, (T + S) (β) = (T + S)α, β = T α + Sα, β = T α, β + Sα, β = = α, T β + α, S β = α, T β + S β. As this holds for any α, (T + S) (β) = T (β) + S (β). As this holds for any β, (T + S) = T + S. (2) Similarly, α, (at ) (β) = (at )(α), β = a T α, β = a α, T β = α, at β. As this holds for any α, (at ) (β) = (at )(β). As this holds for any β, (at ) = at. (3) Here we have α, (T S) (β) = (T S)α, β = T (S(α)), β = S(α), T (β) = α, S (T (β)). This holds for any α, so (T S) (β) = S (T (β)). This holds for any β, so that (T S) = S T. (4) Observe that α, (T ) (β) = T (α), β = β, T (α) = T β, α = α, T β. 4
5 This holds for any α, so (T ) (β) = T (β). This holds for any β, so (T ) = T. (5) If T = S then (T ) = (S ) and by (4) T = S. Thus * is injective. For a given T, T exists, and (T ) = T, so * is also surjective. Since the * operation preserves the relation between matrices and transformations, these properties hold for matrices as well. It is left as an exercise to prove it directly for matrices. Let I be the identity, then I is also the identity matrix, since Iα, β = α, Iβ = α, I β. Definition 3.7 T is conjugate to itself if T = T. When F = R, it is said to be symmetric. When F = C, it is said to be Hermitian. An n n matrix is conjugate to itself if A = A, which happens precisely when A is symmetric if F = R. If F = C then A is said to be Hermitian. If A represents T under an orthonormal basis then A is Hermitian if and only if T is Hermitian. Lemma 3.8 Let T be conjugate to itself and suppose that T α, α = 0 for any α then T = 0. Proof. If F = C then we do not even need T to be conjugate to itself, so it is sufficient to prove the lemma for F = R. One has 0 = T (α + β), α + β = T α, α + T β, β + T α, β + T β, α = = T β, α + T α, β = T α, β + β, T (α) = T α, β + β, T (α) = 2 T α, β, where the last equality is because F = R and the one before that is because T is conjugate to itself. Since T α, β = 0 for any α, β, it follows that T = 0. Theorem 3.9 Let V be a unitary space, then T is conjugate to itself if and only if α V we have T α, α R. This is obviously true only when F = C. Proof. Suppose that T is conjugate to itself. Then for any α we have T α, α = α, T α = α, T α = T α, α, from which it follows that it is indeed a real number. For the converse, reversing the arguments shows that α V α, (T T )(α) = 0, and since F = C it follows that T = T. 5
6 Remark 1 Obviously, T is conjugate to itself if and only if T is conjugate to itself. Definition 3.10 T is unitary if it preserves the inner product, i.e. for any α, β V we have T α, T β = α, β. In a Euclidean space a unitary transformation is called orthogonal. Theorem 3.11 The following are equivalent (1) T is unitary. (2) T preserves the norm i.e. a = T (a) for any a. (3) T T = I. (4) T maps any orthonormal basis to an orthonormal basis. (5) T maps some orthonormal basis to an orthonormal basis. Remark 2 (2) is equivalent to T preserving lengths i.e. T α T β = T (α β) = α β. (3) means that T and T are invertible, and the inverse of one is the other. Clearly this means T T = I as well. Therefore, if T is unitary then T is as well. Proof. If T is unitary, then T α 2 = T α, T α = α, α = α 2, so T preserves the norm. If T preserves the norm, then α, T T α = T α, T α = α, α. Thus 0 = α, T T α α = α, (T T I)α = (T T I)α, α. The proof now follows since T T I is conjugate to itself : (T T I) = T T I. If T T = I and {e 1,..., e n } is an orthonormal basis for V, then e i, e j = δ ij. As T is invertible, {T e 1,..., T e n } has to be a basis. It remains to show that it is orthonormal. T e i, T e j = e i, T T e j = e i, e j = δ ij, so {T e 1,..., T e n } is an orthonormal basis. (4) implies (5), since orthonormal bases exist. Suppose that {e 1,..., e n } and {T e 1,..., e n } are orthonormal bases. Then for α = i a ie i and β = j b je j we have T α, T β = ai T e i, b j T e j = a i T e i, b j T e j = = i a i b j e i, e j = a i b i = α, β. j Definition 3.12 A matrix A, is said to be unitary (orthogonal, if F = R) if A A = I (A t A = I). 6
7 Corollary 3.13 If A represents T with respect to an orthonormal basis, then T is unitary if and only if A is unitary. Proof. If T is unitary then T T = I, and as the basis is orthonormal, A represents T, and thus A A represents I, which gives A A = I. If T is not unitary then T T I and A will not be unitary. ( ) ( ) cos a sin a 1 0 Example : is an orthogonal matrix over R, as well as. sin a cos a 0 1 A product of unitary matrices is also unitary : (T S) T S = S T T S = S IS = S S = I. Proposition 3.14 A is unitary if and only if there exists an inner product space V over F such that A is the transformation matrix of a transformation mapping an orthonoraml basis of V to an orthonormal basis of V. Proof. Let A be unitary, set V = F n with the standard inner product, and define T : V V by T α = Aα. Then T is a unitary linear transformation. Let k = {e 1,..., e n } be the standard basis of V. Thus k 2 = {T e 1,..., T e n } is an orthonormal basis as well. A is the transformation matrix from k to k 2. For the converse, we use the isomorphism that maps a vector to its coordinates vector under the orthonormal basis k = {e 1,..., e n }. Now, suppose that k 2 = {d 1,..., d n } and k are both orthonormal, and A is the transformation matrix from k to k 2. Therefore T as defined above maps k to k 2 ; it is therefore unitary. A represents it under an orthonormal basis, so it is also unitary. Theorem 3.15 Let A M n (F ) and consider its rows as vectors α 1,..., α n F n. Then A is unitary if and only if these rows form an orthonormal basis. Proof. Write A A = (b ij ), then A is unitary if and only if b ij = δ ij. On the other hand, α i, α j = k a ika jk = k a ika kj = b ij. We need α i, α j to equal δ ij and that happens if and only if A is unitary. It follows immediately that A is unitary if and only if its columns are orthonormal basis. We wish to discover when V has an orthonormal basis of eigenvectors of T. Claim 1 If such a basis exists, then T T = T T. Proof. If {e 1,..., e n } is an orthonormal basis of eigenvectors of T then the matrix A which represents T under this basis is diagonal. A is diagonal as well, and represents T. AA represents T T, and A A represents T T, but AA = A A because both are diagonal. Thus T T = T T. 7
8 Definition 3.16 A linear transformation T is said to be normal if T T = T T. Examples : If T is conjugate to itself then it is normal. If it is unitary then it is normal. Lemma 3.17 Let V inner product space over F, T : V V normal. If c is eigenvalue of T with eigenvector u, then c is an eigenvalue of T with the same u. Proof. As T is normal, for any w V we have T w 2 = T w, T w = w, T T w = w, T T w = T w, T w = T w 2. We know that T u = cu and wish to show that T u = cu. Consider T ci. As T is normal, T ci is also normal : (T ci)(t ci) = (T ci)(t ci) = T T ct ct + cci, and (T ci) (T ci) = T T ct ct + cci, which are equal as T and T commute. Thus T ci is normal. It follows that 0 = T u cu = (T ci)u = (T ci) u = (T ci)u = T u cu. This gives T u = cu. Corollary 3.18 If T is normal, then any real eigenvalue of T is an eigenvalue of T. Lemma 3.19 Let T be normal, then any two eigenvectors lying in different eigenspaces are orthogonal. Proof. Suppose that u belong to a and w to b a, then a u, w = au, w = T u, w = u, T w = u, bw = b u, w, in virtue of the previous lemma. As a b this means u, w = 0. Lemma 3.20 Let V be an inner product space over C, with 0 < dimv = n <. Then for any linear transformation T : V V there is an eigenvector u 0. Proof. The (not identically zero) characteristic polynomial splits to linear components and they are all eigenvalues thus there are nonzero eigenvectors. Theorem 3.21 Let V be a unitary space over C and T : V V be normal. Then V admits an orthonormal basis of eigenvectors of T. 8
9 Proof. Let U be the span of all the eigenvectors. This is a subspace of V. Let a 1,..., a k be the eigenvalues of T and V i U the eigenspaces. For any V i we choose an orthonormal basis and define E to be the union of theses bases. Since all the eigenvectors are in E we have U = V i = span(e). Write E = {u 1,..., u l }, where l k, then E spans U. We have to show that E is independent and that l = n. If i = j, then u i, u j = 1 because u i is an elements of an orthonormal basis of some space. If i j and u i, u j are in the same eigenspace V r, then u i, u j = 0 because they are part of an orthonormal basis of V r. If i j and they lie in different eigenspaces, then u i, u j = 0 by a previous lemma. This proves that E is an orthonormal system, thus independent. To see that l = n, write V = U U +, where U + is the orthogonal complement of U. We wish to show that U + = {0}, and we begin by showing that it is T invariant, i.e. T (U + ) U +. Let β U +, then u j, β = 0 for any j. But u j, T (β) = T (u j ), β = xu j, β = x u j, β = 0, where x is the eigenvalue to whose space u j belongs. Thus T (β) U +. Define S : U + U + by S(α) = T (α). If there is α 0, then by a previous lemma S has an eigenvector. It will also be an eigenvector of T, so must lie in U, and so the intersection of U and U + is not {0}, a contradiction. This completes the proof. Corollary 3.22 Let T : V V as before, then V has an orthonormal basis under which T is diagonal. Corollary 3.23 For a normal A M n (C), there is a unitary U M n (C) with U AU = U 1 AU diagonal. In order to find U, one can proceed as follows : (1) calculate the characteristic polynomial det(xi A). (2) find the eigenvalues of A, i.e. the roots of the polynomial. (3) for any eigenvalue find a basis for the eigenspace. (4) for any eigenspace find an orthornormal basis (use Gram Schimdt process). (5) the union of those bases is an orthonormal basis of eigenvectors of V. If the columns of U are (α 1,..., α n ), then the similar matrix is l 1 l 2..., l n 9
10 where l i is the eigenvalue of α i (l 1,..., l n do not have to be distinct). Theorem 3.24 Let V be an inner product space over C or R, T : V V normal. Then (1) T is conjugate to itself if and only if all the eigenvalues of T (in C) are real numbers. (2) T is unitary if and only if all eigenvalues have an absolute value of 1. ( ) 0 1 Example : is unitary (its rows are orthonormal), and has no real 1 0 eigenvalues. But over C is has the eigenvalues i and i, both have a modulus of 1. Proof. Let {e 1,..., e n } be an orthorormal basis of V under which the matrix D of T is diagonal. D is clearly normal. (1) T is conjugate to itself if and only if D = D, which is equivalent to D having only real eigenvalues, and they are the same eigenvalues of T. (2) T is unitary if and only if D is unitary if and only if DD = I if and only if l i l i = 1 i if and only if li = 1 i. Corollary 3.25 Let A be a symmetric matrix over R then A is diagonalizable over R. Proof. A is symmetric thus conjugate to itself thus normal. Therefore all the eigenvalues are real, and the characteristic polynomial decomposes to linear elements. As A is normal it can be diagonalized by a unitary matrix : for any eigenvalue we look at the eigenspace and find an orthonormal basis. Since A and its eigenvalues are real, we can choose the elements of this basis to be real as well. The union of these bases is an orthonormal basis of V. The matrix U whose columns are the vectors of the basis is orthogonal and U 1 AU is diagonal. Corollary 3.26 Let V be a unitary space with dimv = n. Let T : V V, then T is normal if and only if there exist a unitary matrix U and a Hermitian matrix H such that HU = UH = T. Proof. If such H and U exist then HU is normal, because (H = H commutes with U) HU(HU) = HUU H = HH = H 2 = U UHH = U HHU = U H HU = (HU) HU. If T is normal, let {e 1,..., e n } be an orthonormal basis of V under which D, the matrix that represents T, is diagonal. Write l 1 l 2 D =..., l j = r j (cos t j + i sin t j ), r j R +. l n 10
11 Set r 1 H r 2 = cos t 1 + i sin t 1..., U cos t 2 + i sin t 2 = r n.... cos t n + i sin t n Then : H is Hermitian because it is diagonal, and all its eigenvalues are real. U is normal because it is diagonal. It is unitary because all its eigenvalues have a modulus of 1. Clearly H U = U H = D. Now let H, and U be the linear transformations whose matrices under {e 1,..., e n } are H and U respectively. Then HU = UH = T. 4 Bilinear patterns Definition 4.1 Let W and V be vector space over a field F. A bilinear pattern is a function f : V W F with the following properties : f(a + c, b) = f(a, b) + f(c, b), f(a, b + c) = f(a, b) + f(a, c), f(au, v) = f(u, av) = af(u, v). Equivalently, for any v V and w W, f(v, ) : W F and f(, w) : V F are linear functionals. Examples : f(v, w) = 0 for any w, v is a bilinear pattern. If V is a Euclidean space over over R then, : V V F is bilinear. If V is unitary,, is not bilinear because it is not homogeneous unless V = {0}. If V is the dual space of V then f : V V F defined by f(v, w) = w(v) is a bilinear pattern. Fix g W and h V. Then f(w, v) = g(w)h(v) is also a bilinear pattern. Let V = F m, W = F n and A an m n matrix. Then A defines a bilinear pattern on V W by f(v, w) = v t Aw where v t is the transpose of v. This example is in fact the most general, once we fix bases for V and W : Suppose that dimv = m and dimw = n. Let (v 1,..., v m ) and (w 1,..., w n ) be bases. For a bilinear f define the matrix A by A ij = f(v i, w j ). 11
12 Write v = m i=1 x iv i, w = n i=1 y iw i then ( f(v, w) = f x i v i, ) y i w i i j = i,j x i y j f(vi, wj) = = i,j x i y j A ij = (x 1,..., x m )A(y 1,..., y n ) t = v t Aw, where v t = (x 1,..., x m ) and w t = (y 1,..., y n ). Given these two bases, f determines A and A determines f, so the map f A is bijective. If W = V and A = I under the standard basis we get f(u, v) = u t v. Suppose that dimv = m and dimw = n. multiplication by Define addition and scalar (f + g)(a, b) = f(a, b) + g(a, b) and (cf)(a, b) = cf(a, b). Then the collection of bilinear patterns is a vector space, of dimension mn by the isomorphism f A. One can also show it without using matrices : Lemma 4.2 Let V and W be vector space of finite dimension, and {g 1,..., g m }, {h 1,..., h n } bases of V and W respectively. Then g i h j is a bilinear pattern. These mn products give a basis for the vector space of bilinear patterns. Theorem 4.3 Let f be a bilinear pattern on V W and consider its matrix A with respect to the bases {w 1,..., w n } and {v 1,..., v m }. Let P be the transformation matrix from {v 1,..., v m } to {v 1,..., v m} and Q the transformation matrix from {w 1,..., w n } to {w 1,..., w n} then the matrix of f with respect to these new bases is P t AQ. Proof. Let v V, w W, and write their coordinates under {v 1,..., v m} and {w 1,..., w n} as v = (x 1,..., x m ) and w = (y 1,..., y n ). Then the coordinates of v under {v 1,..., v m } are P (x 1,..., x m ), and the coordinates of w under {w 1,..., w n } are Q(y 1,..., y n ), and so f(v, w) = (P (x 1,..., x m )) t AQ(y 1,..., y n ) = (x 1,..., x m ) t (P t AQ)(y 1,..., y n ), so the matrix by the new bases is indeed P t AQ. If V = W, then we call f a linear pattern on V. In such cases, we use the same basis for both vectors. Thus, if the matrix of f under {v 1,..., v n } is A, and under {v 1,..., v n} it is A, then A = P t AP, where P is the transformation matrix from {v 1,..., v n } to {v 1,..., v n}. 12
13 Definition 4.4 Two matrices A and B are congruent if there exists an invertible matrix P such that B = P t AP. This defines an equivalence relation, since A = I t AI, and if B = P t AP then A = (P t ) 1BP 1 = (P 1) t BP 1. If C = Q t BQ then C = Q t BQ = Q t P t AP Q = (P Q) t A(P Q), and P Q is invertible. Lemma 4.5 Two n n matrices are congruent if and only if they represent the same bilinear pattern over V. Proof. Immediate. Theorem 4.6 Let f be a bilinear pattern on V W with dimv = m and dimw = n. Choose bases {v 1,..., v m } and {w 1,..., w n } and 0 r min{m, n}. { 1 i = j r (1) f(v i, w j ) =. This gives something of the sort of a diagonal 0 otherwise matrix. 13
DSGA 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 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 informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
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
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 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 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 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 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 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 informationNumerical Linear Algebra
University of Alabama at Birmingham Department of Mathematics Numerical Linear Algebra Lecture Notes for MA 660 (1997 2014) Dr Nikolai Chernov April 2014 Chapter 0 Review of Linear Algebra 0.1 Matrices
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 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 informationReal symmetric matrices/1. 1 Eigenvalues and eigenvectors
Real symmetric matrices 1 Eigenvalues and eigenvectors We use the convention that vectors are row vectors and matrices act on the right. Let A be a square matrix with entries in a field F; suppose that
More informationLinear Algebra 2 Spectral Notes
Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex
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 informationSpectral Theorem for Selfadjoint Linear Operators
Notes for the undergraduate lecture by David Adams. (These are the notes I would write if I was teaching a course on this topic. I have included more material than I will cover in the 45 minute lecture;
More informationThe Spectral Theorem for normal linear maps
MAT067 University of California, Davis Winter 2007 The Spectral Theorem for normal linear maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 14, 2007) In this section we come back to the question
More informationDiagonalizing Matrices
Diagonalizing Matrices Massoud Malek A A Let A = A k be an n n nonsingular matrix and let B = A = [B, B,, B k,, B n ] Then A n A B = A A 0 0 A k [B, B,, B k,, B n ] = 0 0 = I n 0 A n Notice that A i B
More informationSpectral Theorems in Euclidean and Hermitian Spaces
Chapter 10 Spectral Theorems in Euclidean and Hermitian Spaces 10.1 Normal Linear Maps Let E be a real Euclidean space (or a complex Hermitian space) with inner product u, v 7! hu, vi. In the real Euclidean
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 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 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 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 informationSpectral Theorems in Euclidean and Hermitian Spaces
Chapter 9 Spectral Theorems in Euclidean and Hermitian Spaces 9.1 Normal Linear Maps Let E be a real Euclidean space (or a complex Hermitian space) with inner product u, v u, v. In the real Euclidean case,
More informationMATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2.
MATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2. Diagonalization Let L be a linear operator on a finitedimensional vector space V. Then the following conditions are equivalent:
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 informationVector Spaces and Linear Transformations
Vector Spaces and Linear Transformations Wei Shi, Jinan University 2017.11.1 1 / 18 Definition (Field) A field F = {F, +, } is an algebraic structure formed by a set F, and closed under binary operations
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 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 informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
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 informationLecture notes on Quantum Computing. Chapter 1 Mathematical Background
Lecture notes on Quantum Computing Chapter 1 Mathematical Background Vector states of a quantum system with n physical states are represented by unique vectors in C n, the set of n 1 column vectors 1 For
More informationThe Singular Value Decomposition
The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will
More information22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices
m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix
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 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 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 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 informationReview of linear algebra
Review of linear algebra 1 Vectors and matrices We will just touch very briefly on certain aspects of linear algebra, most of which should be familiar. Recall that we deal with vectors, i.e. elements of
More information2.2. Show that U 0 is a vector space. For each α 0 in F, show by example that U α does not satisfy closure.
Hints for Exercises 1.3. This diagram says that f α = β g. I will prove f injective g injective. You should show g injective f injective. Assume f is injective. Now suppose g(x) = g(y) for some x, y A.
More informationA PRIMER ON SESQUILINEAR FORMS
A PRIMER ON SESQUILINEAR FORMS BRIAN OSSERMAN This is an alternative presentation of most of the material from 8., 8.2, 8.3, 8.4, 8.5 and 8.8 of Artin s book. Any terminology (such as sesquilinear form
More informationChapter 1 Vector Spaces
Chapter 1 Vector Spaces PerOlof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field
More informationChapter 6: Orthogonality
Chapter 6: Orthogonality (Last Updated: November 7, 7) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved around.. Inner products
More informationis an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent
Lecture 4. GModules PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 4. The categories of Gmodules, mostly for finite groups, and a recipe for finding every irreducible Gmodule of a
More informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tuberlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
More informationMATH SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER Given vector spaces V and W, V W is the vector space given by
MATH 110  SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER 2009 GSI: SANTIAGO CAÑEZ 1. Given vector spaces V and W, V W is the vector space given by V W = {(v, w) v V and w W }, with addition and scalar
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 informationSchur s Triangularization Theorem. Math 422
Schur s Triangularization Theorem Math 4 The characteristic polynomial p (t) of a square complex matrix A splits as a product of linear factors of the form (t λ) m Of course, finding these factors is a
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 information1. Foundations of Numerics from Advanced Mathematics. Linear Algebra
Foundations of Numerics from Advanced Mathematics Linear Algebra Linear Algebra, October 23, 22 Linear Algebra Mathematical Structures a mathematical structure consists of one or several sets and one or
More informationSymmetric and selfadjoint matrices
Symmetric and selfadjoint matrices A matrix A in M n (F) is called symmetric if A T = A, ie A ij = A ji for each i, j; and selfadjoint if A = A, ie A ij = A ji or each i, j Note for A in M n (R) that
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 informationLecture 10: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11)
Lecture 1: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11) The eigenvalue problem, Ax= λ x, occurs in many, many contexts: classical mechanics, quantum mechanics, optics 22 Eigenvectors and
More information1 Linear Algebra Problems
Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skewhermitian if A = A and
More informationLECTURE VI: SELFADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY
LECTURE VI: SELFADJOINT AND UNITARY OPERATORS MAT 204  FALL 2006 PRINCETON UNIVERSITY ALFONSO SORRENTINO 1 Adjoint of a linear operator Note: In these notes, V will denote a ndimensional euclidean vector
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 informationEigenvalues and Eigenvectors
/88 ChiaPing Chen Department of Computer Science and Engineering National Sun Yatsen University Linear Algebra Eigenvalue Problem /88 Eigenvalue Equation By definition, the eigenvalue equation for matrix
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 informationNOTES ON BILINEAR FORMS
NOTES ON BILINEAR FORMS PARAMESWARAN SANKARAN These notes are intended as a supplement to the talk given by the author at the IMSc Outreach Programme Enriching Collegiate Education2015. Symmetric bilinear
More informationW2 ) = dim(w 1 )+ dim(w 2 ) for any two finite dimensional subspaces W 1, W 2 of V.
MA322 Sathaye Final Preparations Spring 2017 The final MA 322 exams will be given as described in the course web site (following the Registrar s listing. You should check and verify that you do not have
More informationMathematics 1. Part II: Linear Algebra. Exercises and problems
Bachelor Degree in Informatics Engineering Barcelona School of Informatics Mathematics Part II: Linear Algebra Eercises and problems February 5 Departament de Matemàtica Aplicada Universitat Politècnica
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
More informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
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 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 informationAlgebra II. Paulius Drungilas and Jonas Jankauskas
Algebra II Paulius Drungilas and Jonas Jankauskas Contents 1. Quadratic forms 3 What is quadratic form? 3 Change of variables. 3 Equivalence of quadratic forms. 4 Canonical form. 4 Normal form. 7 Positive
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 informationEcon 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms. 1 Diagonalization and Change of Basis
Econ 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms De La Fuente notes that, if an n n matrix has n distinct eigenvalues, it can be diagonalized. In this supplement, we will provide
More informationOctober 25, 2013 INNER PRODUCT SPACES
October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal
More informationMath113: Linear Algebra. Beifang Chen
Math3: Linear Algebra Beifang Chen Spring 26 Contents Systems of Linear Equations 3 Systems of Linear Equations 3 Linear Systems 3 2 Geometric Interpretation 3 3 Matrices of Linear Systems 4 4 Elementary
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 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 informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
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 informationHomework 11 Solutions. Math 110, Fall 2013.
Homework 11 Solutions Math 110, Fall 2013 1 a) Suppose that T were selfadjoint Then, the Spectral Theorem tells us that there would exist an orthonormal basis of P 2 (R), (p 1, p 2, p 3 ), consisting
More informationPart 1a: Inner product, Orthogonality, Vector/Matrix norm
Part 1a: Inner product, Orthogonality, Vector/Matrix norm September 19, 2018 Numerical Linear Algebra Part 1a September 19, 2018 1 / 16 1. Inner product on a linear space V over the number field F A map,
More informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
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 System Theory
Linear System Theory Wonhee Kim Lecture 4 Apr. 4, 2018 1 / 40 Recap Vector space, linear space, linear vector space Subspace Linearly independence and dependence Dimension, Basis, Change of Basis 2 / 40
More informationhomogeneous 71 hyperplane 10 hyperplane 34 hyperplane 69 identity map 171 identity map 186 identity map 206 identity matrix 110 identity matrix 45
address 12 adjoint matrix 118 alternating 112 alternating 203 angle 159 angle 33 angle 60 area 120 associative 180 augmented matrix 11 axes 5 Axiom of Choice 153 basis 178 basis 210 basis 74 basis test
More informationMATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (inclass portion) January 22, 2003
MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (inclass portion) January 22, 2003 1. True or False (28 points, 2 each) T or F If V is a vector space
More informationMAT 445/ INTRODUCTION TO REPRESENTATION THEORY
MAT 445/1196  INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups  Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations
More informationCategories and Quantum Informatics: Hilbert spaces
Categories and Quantum Informatics: Hilbert spaces Chris Heunen Spring 2018 We introduce our main example category Hilb by recalling in some detail the mathematical formalism that underlies quantum theory:
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 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 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 information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationAlgebra I Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.701 Algebra I Fall 007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.701 007 Geometry of the Special Unitary
More informationLecture 8 : Eigenvalues and Eigenvectors
CPS290: Algorithmic Foundations of Data Science February 24, 2017 Lecture 8 : Eigenvalues and Eigenvectors Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Hermitian Matrices It is simpler to begin with
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 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 583A REVIEW SESSION #1
MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),
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 information4.2. ORTHOGONALITY 161
4.2. ORTHOGONALITY 161 Definition 4.2.9 An affine space (E, E ) is a Euclidean affine space iff its underlying vector space E is a Euclidean vector space. Given any two points a, b E, we define the distance
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 Final December 2006 C. Robinson
Math 2851 Final December 2006 C. Robinson 2 5 8 5 1 2 01 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the
More informationFORMS ON INNER PRODUCT SPACES
FORMS ON INNER PRODUCT SPACES MARIA INFUSINO PROSEMINAR ON LINEAR ALGEBRA WS2016/2017 UNIVERSITY OF KONSTANZ Abstract. This note aims to give an introduction on forms on inner product spaces and their
More informationRepresentation Theory
Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim
More informationUniversity of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm
University of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm Name: The proctor will let you read the following conditions
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 information