Linear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.
|
|
- Clara Shelton
- 5 years ago
- Views:
Transcription
1 Linear Algebra M1 - FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation: Oriol Caño, Josep Elgueta
2 5 Matrices, systems of linear equations and determinants 51 Matrices: basic operations and row echelon form Review of matrix algebra 1 2 Scalars Matrices By a field of scalars K we will mean a set of numbers with two operations (addition and multiplication) such that - the usual properties are satisfied (commutative, associative, distributive, identity elements) - they are invertible (we can subtract and divide) Examples: R, Q, Z p, C Let m, n 1 be integers A m n matrix with elements in the field K consists of mn elements of K arranged in a table with m rows and n columns We will denote by a ij the element in the ith row and jth column A generic matrix is represented by: a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn We will also use the notation A = (a ij ) m n The set of all m n matrices will be denoted by M m n (K) 3 4
3 Types of matrices A matrix of type 1 n is called a row matrix A matrix of type m 1 is called a column matrix The null matrix O m,n (or simply O) is the matrix of type m n with all elements equal to 0 A matrix of type n n is called a square matrix The set of all square n n matrices with elements in K is denoted by M n (K) A square matrix (a ij ) n n is upper triangular if aij = 0 for all i > j lower triangular if aij = 0 for all i < j diagonal if it is both upper and lower triangular The matrix Diag(λ 1,, λ n ) is the diagonal matrix (d ij ) n n with d ii = λ i for all i The identity matrix I n is the diagonal matrix Diag(1, 1,, 1) Matrix addition Let A, B M m n (K) with A = (a ij ) and B = (b ij ) Their sum is the matrix A + B = (c ij ) M m n (K) defined by c ij = a ij + b ij Properties If A, B, C M m n (K), the following holds: (Associative) (A + B) + C = A + (B + C) (Commutative) A + B = B + A (Identity element) A + O = O + A = A (Additive inverse) There exists a matrix B such that A + B = B + A = O (This matrix B is denoted A) 5 6 Scalar multiplication Let A M m n (K) with A = (a ij ) and let λ K be a scalar The multiplication of A by the scalar λ is the matrix λa = (b ij ) M m n (K) defined by b ij = λa ij Properties If λ, µ K and A, B M m n (K), the following holds: (Pseudo-associative) λ(µa) = (λµ)a (Distributive 1) λ(a + B) = λa + λb (Distributive 2) (λ + µ)a = λa + µa (Identity) 1A = A Transposition Let A = (a ij ) m n M m n (K) Its transpose is the matrix A t = (b ij ) n m M(K) n m defined by b ij = a ji Clearly (A t ) t = A A square matrix A is symmetric if A t = A skew-symmetric if A t = A Note that ( 1)A = A 7 8
4 Matrix multiplication Let A = (a ij ) m n M m n (K) and B = (b ij ) n p M n p (K) Their product is the matrix AB = (c ij ) m p M m p (K) with c ij = Observations n a ik b kj = a i1 b 1j + a i2 b 2j + + a in b nj k=1 The product of two matrices may not exist BA may not exist even when AB exists If AB and BA are defined, it may be AB BA Multiplication is an internal map in M n (K) Properties of matrix multiplication For matrices A, B, C the following holds (wherever the operations are defined): (Associative) (AB)C = A(BC) (Distributive) A(B + C) = AB + AC and (A + B)C = AC + BC (Identity element) IA = A = AI, where I is the identity matrix of the required type (Relation with the transpose) (AB) t = B t A t If A M n (K), we will denote by A k the product AA A (ie, A 2 = AA, A 3 = AAA, etc) 9 10 Inverse matrix Let A, B M n (K) We say that B is the inverse matrix of A when AB = BA = I n If this holds we say that A is invertible and we denote by A 1 the inverse matrix Observations If the inverse exists, it is unique Not all matrices have an inverse Invertible matrices have no zero rows or columns Properties of the inverse matrix If A and B are invertible matrices of the same type and λ is a nonzero scalar, the following holds: the matrix A 1 is invertible and (A 1 ) 1 = A the matrix A k is invertible and (A k ) 1 = (A 1 ) k the matrix λa is invertible and (λa) 1 = (λ) 1 A 1 the matrix A t is invertible and (A t ) 1 = (A 1 ) t the product AB is invertible and (AB) 1 = B 1 A
5 Elementary row operations Let A M m n (K) Elementary row operations and row echelon form An elementary row operation of A consists of one of the following three operations: (I) switching two rows of A (II) multiplying a row of A by a nonzero scalar (III) adding to one row of A the result of multiplying another row by a nonzero scalar A matrix is (row) elementary if it can be obtained from an identity matrix by performing only one elementary row operation Matrix equivalence Row echelon form Let T be an elementary row operation and let M M m n (K) The result of applying the operation T to the matrix M is EM, where E is the elementary matrix obtained by applying T to the identity I m A matrix B is (row) equivalent to a matrix A if B can be obtained from A by a finite sequence of elementary row operations So, if B is equivalent to A we can write B = E r E r 1 E 2 E 1 A, where all E i are elementary matrices A matrix is in (row) echelon form if it satisfies the following: - if it has a zero row (all entries equal to zero), all the rows below it are also zero - in each nonzero row, the first nonzero element is equal to 1 (it is called the leading coefficient or pivot of the row) - the pivot of a row is always to the right of the pivot on the previous row Every matrix is equivalent to a matrix in echelon form The rank of a matrix A is the number of nonzero rows of any matrix in echelon form equivalent to A 15 16
6 Application: computing the inverse (I) Application: computing the inverse (II) Lemma If E is an elementary matrix, then E is invertible and its inverse E 1 is also an elementary matrix Proof: (I) If B is an elementary matrix corresponding to a type (I) row operation (switching rows i and j), we have BB = I (II) If C λ is the elementary matrix corresponding to a type (II) row operation (multiplying a row by λ 0), we have C λ C λ 1 = I = C λ 1C λ (II) If D k is the elementary matrix corresponding to a type (III) row operation (adding to row i row j multiplied by k), we have D k D k = I = D k D k Let A M n (K) and let M be a matrix in row echelon form equivalent to A Then A is invertible if and only if all the elements in the diagonal of M are equal to 1 Corollary A matrix A M n (K) is invertible if and only if the rank of A is n Gauss-Jordan method for computing the inverse Let A M n (K) The proof of the previous theorem implies that if I n = E r E 2 E 1 A, then A 1 = E r E 2 E 1 Given A, its inverse A 1, if it exists, can be found as follows: Start with the matrix (A I n ) Apply row operations to (A I n ), until we get a matrix of the form (I n B) If we succeed, A 1 = B Otherwise, A is not invertible 19
7 Systems of linear equations 5 Matrices, systems of linear equations and determinants 52 Systems of linear equations A linear equation in the variables x 1,, x n is an expression of the form a 1 x 1 + a 2 x a n x n = b, where a 1,, a n, b belong to the field of scalars K A solution is (s 1,, s n ) K n such that a 1 s 1 + a 2 s a n s n = b (Obs A linear equation can have no solution, one, or more than one, even infinite, solutions) Systems of linear equations Solutions of a system A system of linear equations is a set of linear equations (all of them with the same variables x 1,, x n ) The general form of a system of linear equations is: a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a m1 x 1 + a m2 x a mn x n = b m A solution to the system is an n-tuple (s 1,, s n ) K n that satisfies all the equations We will say that a system is inconsistent if it has no solution determined if it has only one solution underdetermined if it has more than one solution The general solution of a system is the set of all its solutions Two systems are equivalent if they have the same set of solutions 22 23
8 Equivalent systems Two systems with the same equations but sorted differently are equivalent And if in a system we multiply an equation by a (nonzero) scalar, or we add to an equation a multiple of another one the resulting system is equivalent to the first one Matrix form of a system Given the system a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a m1 x 1 + a m2 x a mn x n = b m the system matrix and the matrices of variables and of constant terms are a 11 a 12 a 1n x 1 b 1 a 21 a 22 a 2n A = x = x 2 b = b 2 a m1 a m2 a mn x n b m We can write the system as a product of matrices: Ax = b Augmented matrix The augmented matrix is the matrix (A b), ie, a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 (A b) = a m1 a m2 a mn b m Obs If row operations are applied to the augmented matrix of a system, the resulting system is equivalent to the first one So, every system of linear equations is equivalent to another one whose augmented matrix is in echelon form Solving a system in row echelon form The row echelon form of a consistent system looks like x 1 + c 12 x 2 + c 13 x c 1r x r + + c 1n x n = d 1 x 2 + c 23 x c 2r x r + + c 2n x n = d 2 x r + + c rn x n = d r (rearranging the variables if needed) Variables x 1,, x r are called basic and the others are called free The system is solved upwards as follows The basic variable x r is isolated in terms of the free variables: x r = d r c r,r+1 x r+1 c rn x n 26 Now we can isolate x r 1 in terms of x r and the rest of the free variables, etc 27
9 General solution of a system in row echelon form In a system in row echelon form we can express all the basic variables in terms of the free ones (and the coefficients): x 1 = f 1 + e 1,r+1 x r e 1,n x n x 2 = f 2 + e 2,r+1 x r e 2,n x n x r = f r + e r,r+1 x r e r,n x n This is the general solution of the system Obs For each assignment of a scalar value to each of the free variables x r+1,, x n we get a particular solution of the system We say that the system has n r degrees of freedom 28 Parametric form of the general solution If the general solution of a system is x 1 = f 1 + e 1,r+1 x r e 1,n x n x 2 = f 2 + e 2,r+1 x r e 2,n x n x r = f r + e r,r+1 x r e r,n x n the parametric form of the solution is the expression x 1 f 1 e 1,r+1 x 2 f 2 e 2,r+1 x r = f r + x r+1 e r,r x n x r x n 0 0 e 1,n e 2,n e r,n Discussing a system: the Rouché-Frobenius Homogeneous systems Let be a system of linear equations with associated matrix A M m n (K) and augmented matrix (A b) Let r, r be the ranks of A, (A b) respectively Then, if r < r, the system is inconsistent if r = r = n, the system is determined if r = r < n, the system is underdetermined with n r degrees of freedom The rank of the associated matrix of a consistent linear system is called the rank of the system A system of linear equations is homogeneous if all the constant terms are equal to 0 Obs A homogeneous system is always consistent (since we have the trivial solution x 1 = = x n = 0) Corollary Let A be the associated matrix of a homogeneous system in n variables; let r be the rank of A Then if r = n, the system is determined and the only solution is the trivial one if r < n, the system is underdetermined and it has non-trivial solutions 30 31
10 Solving the system: Gaussian elimination To find the general solution of any system of linear equations we proceed as follows: 1 Find the augmented matrix (A b) 2 Find a matrix in row echelon form M equivalent to (A b) 3 Apply the Rouché-Frobenius theorem to determine whether the system is consistent or not 4 In the case the system is consistent, find the general solution from the equivalent system with augmented matrix M 32
11 5 Matrices, systems of linear equations and determinants 53 Determinants Definition of determinant Let A = (a ij ) M n (K) A minor of A is any matrix obtained by deleting a certain number of rows and the same number of columns of A The minor associated to the element a ij is the matrix A ij obtained by deleting row i and column j from A The minor A ij is an (n 1) (n 1) square matrix The determinant of A is recursively defined by - if n = 1, then det(a) = a 11 - if n 2, then det(a) = a 11 det(a 11 ) a 12 det(a 12 ) + + ( 1) 1+j a 1j det(a 1j ) + + ( 1) n+1 a 1n det(a 1n ) The cofactor of the element a ij is ( 1) i+j det(a ij ) Computing the determinant (Instead of det(a), sometimes we write A ) Matrices 2 2 and 3 3: a b c d = a det((d)) b det((c)) = ad bc a b c d e f = a e f g h i h i b d f g i + c d e g h = a(ei fh) b(di fg) + c(dh eg) = aei + cdh + bfg ceg afh bdi One can prove by induction: If A has a zero row or column then det(a) = 0 If A = Diag(a 1, a 2,, a n ), then det(a) = a 1 a 2 a n Let A M n (K) and i, j [n] Then det(a) = n a ik ( 1) i+k det(a ik ) k=1 (Expansion of the determinant along row i) det(a) = n a kj ( 1) k+j det(a kj ) k=1 (Expansion of the determinant along column j) 35 36
12 Determinants and row operations Let A, B M n (K) If B is the matrix obtained from A switching two rows, then det(b) = det(a) (row operation of type (I)) multiplying the i-th row of A by λ, then det(b) = λ det(a) (row operation of type (II)) adding to a row a multiple of another one, then det(b) = det(a) (row operation of type (III)) Corollary If M is obtained from A by performing row operations, det(m) = K det(a), where K 0 Characterization of invertible matrices A matrix A M n (K) is invertible if and only if det(a) 0 Corollary A matrix A M n (K) has rank n if and only if det(a) 0 Let A M m n (K) The rank of A is r if and only if the largest minor of A with nonzero determinant has type r r So, if A and M are equivalent matrices, then det(a) 0 det(m) Determinants and operations with matrices If A, B M n (K), then det(ab) = det(a) det(b) det(a t ) = det(a) if A is invertible, det(a 1 ) = det(a) 1 However, in general det(a + B) det(a) + det(b) 39
13 6 Vector spaces Properties Addition in R n satisfies the following properties, for all u, v, w R n : s1) (associative) u + (v + w) = (u + v) + w s2) (commutative) u + v = v + u s3) (identity element) u + 0 = u where 0 = (0 0 0) t s4) (additive inverse) for all u there exists u such that u + u = 0 The scalar multiplication in R n satisfies the following properties, for all u, v R n and λ, µ R: p1) λ(µu) = (λµ)u p2) λ(u + v) = λu + λv p3) (λ + µ)u = λu + µu p4) 1u = u (All properties hold because they hold in R and the operations are defined componentwise) R n and its operations x 1 R n x 2 = { : x i R, 1 i n} x n x 1 y 1 x 2 Let x = and y = y 2 be elements of Rn and λ R x n y n Addition in R n : Scalar Multiplication in R n : x 1 + y 1 λx 1 x 2 + y 2 x + y = λx 2 λx = x n + y n λx n (ie, both operations are defined componentwise) 62 Vector spaces A vector space over a field K consists of 1 a non empty set E 2 a map E E E (addition +) and 3 a map K E E (scalar multiplication ) such that the following holds for all u, v, w E and all λ, µ K: e1) (associative) u + (v + w) = (u + v) + w e2) (commutative) u + v = v + u e3) (identity element) there exists a unique element 0 E E such that u + 0 E = u e4) (additive inverse) for each u E there exists a unique u E such that u + u = 0 E e5) λ(µu) = (λµ)u e6) λ(u + v) = λu + λv e7) (λ + µ)u = λu + µu e8) 1u = u, where 1 is the identity element of the product in K 41 43
14 Some examples of vector spaces R n Z n 2 : strings of n bits Addition is bit by bit: eg, Scalar multiplication: 0u = 0 Z n 2 and 1u = u M m n (K) (the m n matrices with entries in the field K) Upper triangular matrices in M n (R) P(R): the set of polynomials with coefficients in R P d (R): the polynomials with degree at most d with coefficients in R The trivial vector space with only the zero vector: {0 E } The solutions of a homogeneous system of linear equations Properties If v E and λ is a scalar, the following holds: 0v = 0 E λ 0 E = 0 E If λ v = 0 E, then λ = 0 or v = 0 E The additive inverse of v is ( 1)v; usually we will write v Vector subspaces and linear combinations A subset S E is a vector subspace (VS) if (s1) S (s2) for all u, v S, u + v S (s3) for all u S and all λ K, λu S The vector 0 E belongs to all vector subspaces Some examples of vector subspaces P d (R) is a vector subspace of the vector space P(R) Upper triangular matrices of M n (R) are a VS of M n (R) The solutions of a homogeneous system of linear equations in n variables and coefficients in K is a VS of K n Intersection of subspaces Lemma If S and S are vector subspaces of E, then S S is also a subspace of E The union of vector subspaces is not a vector subspace in general, as in the case ( ) ( ) x S = { : x R}, and S x = { : x R}; x x ( ) ( ) 1 2 We have + S S 1 2 Linear combination Given vectors u 1,, u k in E, a linear combination of u 1,, u k is any vector of the form λ 1 u λ k u k, where λ 1,, λ k are scalars 46 47
15 Subspaces spanned by some vectors Let u 1,, u k be vectors of E The subspace spanned by u 1,, u k is the set u 1,, u k = {λ 1 u 1 + λ 2 u λ k u k : λ 1,, λ k K}, ie, the set of all linear combinations of u 1,, u k Proposition The subspace u 1,, u k is a vector subspace Moreover, it is the smallest subspace that contains u 1,, u k If a space S can be written as S = u 1,, u l, we will say that {u 1,, u l } is a spanning set of S The spanning set of a space is not unique Observe that v is a linear combination of u 1,, u k if and only if v u 1,, u k Examples of u 1,, u l R n 0 =, 1,, The matrix space M m n (R) is spanned by the matrices M ij that have all their entries equal to 0, except the (i, j) entry, which is equal to 1, 1 i n and 1 j m For example, M 2 (R) = M 11, M 12, M 21, M 22, where M 11 = ( ) 1 0, M = P d (R) = 1, x,, x d ( ) 0 1, M = (These examples can be generalized to any field K) ( ) 0 0, M = ( ) Linear independence Let u 1,, u k E The equation To span the upper triangular matrices, it is enough to use the previous matrices M ij with i j A subspace whose vectors are given in terms of parameters {a + (b a)x + (c b)x 2 + (a c)x 3 : a, b, c R} ={a(1 x + x 3 ) + b(x x 2 ) + c(x 2 x 3 ) : a, b, c R} = 1 x + x 3, x x 2, x 2 x 3 λ 1 u 1 + λ 2 u λ k u k = 0 E always has the solution λ 1 = = λ k = 0 If this is the only solution we will say that the vectors u 1,, u k are linearly independent (LI) If there is a solution with some λ i 0, we will say that the vectors are linearly dependent (LD) (We will also say that the set {u 1,, u k } is LI or LD, resp) Examples: The vector 0 E is linearly dependent Given a vector u 0 E, the vector u is linearly independent If u is any given vector and λ is a scalar, {u, λu} is LD 50 51
16 To determine if a set of vectors u 1, u 2,, u k of R n is linearly independent we proceed as follows: (1) we form a matrix A with the given vectors, putting them in columns (2) we compute the rank r of A (3) if r = k, then the k vectors are LI if r < k, then they are LD; if we have computed the rank putting the matrix A in row echelon form, then the vectors that correspond to the columns with a pivot are a maximal LI subset; if we have computed the rank by minors, the vectors that correspond to the columns of the largest minor of A with nonzero determinant are a maximal LI subset In general, to determine if a set of vectors u 1, u 2,, u k of a vector K-space E is linearly independent we proceed as follows: (1) from the vector equation λ 1 u 1 + λ 2 u λ k u k = 0 E we obtain a homogeneous system with unknowns λ 1, λ 2,, λ k (2) we discuss the system, if it is determined, the vectors u1, u 2,, u k are LI underdetermined, the vectors u1, u 2,, u k are LD Properties Let S = {u 1,, u k } be a set of vectors of a K-vector space E If 0 E is in S, then u 1,, u k are LD If u 1,, u k are LI, then 0 E is not in S If u 1,, u k are LI, every subset of S is LI If u 1,, u k are LD, every set that contains S is LD If u 1,, u k are LD and u 1 is a linear combination of the other vectors in S, then Characterizations A set of vectors S is LD if, and only if, there exists a vector v in S that is a linear combination of the other vectors in S Corollary Let v E If u 1,, u k are LI, then v, u 1,, u k are LI if, and only if, v u 1,, u k u 1, u 2,, u k = u 2,, u k 54 55
17 65 Bases and dimension Let E be a K-vector space A set of vectors B = {b 1, b 2,, b n } is a basis of E if (b1) B is linearly independent (b2) E = b 1, b 2,, b n, ie, b 1, b 2,, b n span E The canonical basis of K n 0 is {, 1,, 0 } of M m n (K) is given by the mn matrices M ij that have all the entries equal to 0 except the (i, j) entry, which is equal to 1 of P d (K) is {1, x, x 2,, x d } ({x d, x d 1,, 1} is also a canonical basis; we will need to specify which one we use) 56 Let B = {b 1,, b n } be a basis of E Proposition Each vector of E can be written in a unique way as a linear combination of the vectors in B Let v E If v = α 1 b α n b n, we say that α 1 v B = α n is the vector of coordinates of v in the basis B Proposition Let {u 1,, u k } be a set of vectors of E that are LI Then k n Corollary Every basis of E has n elements 57 Dimension The cardinal of any basis of a vector space E (or of a VS) will be called the dimension of the space, denoted by dim(e) The dimensions of the spaces that we will mostly work with are: dim(k n ) = n, dim(m m n (K)) = nm, and dim(p d (K)) = d + 1 The dimension of the subspace {0 E } is 0 Let E be of dimension n and let W = {w 1,, w n } be a subset of E if W is a LI set, then W is a basis of E if W spans E, then W is a basis of E If S is a subspace of E then dim(s) dim(e) if dim(s) = dim(e), S = E Change of basis Let B = {b 1,, b n } and B = {b 1,, b n} be two bases of a K-vector space E Let u be a vector of E We look for the relationship between the coordinate vectors u B and u B We call change-of-basis matrix from B to B the matrix whose columns are the vectors of coordinates (b 1 ) B,, (b n ) B It is denoted by P B B PB B = (b 1 ) B (b 2 ) B (b n ) B Then u B = P B B u B PB B = ( PB B )
18 7 Linear Maps Properties Let f : E F be a linear map Then f (0 E ) = 0 F f ( λ i v i ) = λ i f (v i ), for all u E if S is a subspace of E, f (S) is a subspace of F if S is a subspace of F, f 1 (S ) is a subspace of E Proposition Let B = {b 1,, b n } be a basis of E Then f is uniquely determined by f (b 1 ),, f (b n ) ie, the image of any vector u E can be obtained from the image of a basis: if u = α 1 b α n b n, then f (u) = α 1 f (b 1 ) + + α n f (b n ) Corollary If S = v 1,, v k is a subspace of E, then f (S) = f (v 1 ),, f (v k ) Definitions, examples and properties Let E and F be two K-vector spaces A map f : E F is linear if: (a1) for all u, v E, f (u + v) = f (u) + f (v) (a2) for all u E and all λ K, f (λu) = λf (u) If E = F, we say that f is an endomorphism Examples Trivial map f : E F given by f (u) = 0 F, u E, is linear Identity map I E : E E given by I E (u) = u, u E, is linear Let A M m n (K) The map f : K n K m given by f (v) = Av, v K n, is linear The following is a non-linear map (( )) a b f : M 2 2 (R) P 2 (R), f = x 2 (a+d)x+(2c b) The map f : R 2 R 2, f ( x y c d ) ( x = 2 y 2 ) is not linear x + y Let B = {b 1,, b n } be a basis of E, W a basis of F and m the dimension of F The matrix associated to f in the bases B and W is the matrix in M m n (K) whose columns are the coordinates in the basis W of the images of the vectors in the basis B It is denoted by M B W (f) MW B (f ) = f (b 1 ) W f (b 2 ) W f (b n ) W M m n(k) To find the vector of coordinates of the image of a vector u E we just need to compute the matrix product f (u) W = M B W (f )u B 61 63
19 72 Characterization of linear maps A linear bijective map is called an isomorphism Characterization of the type of linear map Let f : E f be a linear map and let M be the matrix associated to f in given bases of E and of F f is injective rank(m) = dim(e) f is surjective rank(m) = dim(f ) f is an isomorphism rank(m) = dim(e) = dim(f ) If E and F have the same dimension, then f is an isomorphism f is injective f is surjective 73 Composition of linear maps Let E, F and G be vector spaces with bases B, W and V, respectively Proposition If f : E F and g : F G are linear maps, the composition g f : E G is also a linear map and Proposition M B V (g f ) = MW V (g)mb W (f ) If f : E F is an isomorphism, the inverse map f 1 : F E is also linear and MB W (f 1 ) = (MW B (f )) Change of basis We look for the change to the matrix associated to a linear map when we change the basis in the source and/or the target space Let f : E F be a linear map, B and B bases of E, and W and W bases of F E B I E P B B E B f M B W (f ) f F W P W W I F F W M B W (f ) f = I F f I E M B W (f ) = P W W MW B (f ) PB B 66
20 The problem of diagonalization Let f : E E be an endomorphism Is there any basis B of E in which the matrix M B (f ) is simple? More concretely, diagonal? 8 Diagonalization Def An endomorphism f : E E is diagonalizable if there exists a basis B of E such that M B (f ) is diagonal Obs The matrix M B (f ) may be non-diagonal, but if we know that the endomorphism f diagonalizes in another basis B, then the matrix (PB B ) 1 M B (f )PB B is diagonal So, being diagonalizable is equivalent to the existence of an invertible matrix P such that P 1 M B (f )P is diagonal Eigenvalues and eigenvectors Def The scalar λ is an eigenvalue of an endomorphism f if there exists a vector v 0 E such that f (v) = λv All vectors v 0 E such that f (v) = λv are called eigenvectors of eigenvalue λ The endomorphism f : E E diagonalizes if and only if there is a basis of E completely made of eigenvectors of f 69 Computing eigenvalues Let M be the matrix associated to f : E E in a basis B Def The characteristic polynomial of the endomorphism f is p f (x) = det(m xi n ) The eigenvalues of f are the roots of the characteristic polynomial The algebraic multiplicity of an eigenvalue λ is the multiplicity of λ as a root of p f (x) and it is denoted by m λ The equation p f (x) = 0 is called the characteristic equation The characteristic polynomial does not depend on the basis in which we compute the matrix M 70
21 Eigenspaces Characterization of diagonalizable endomorphisms Let λ be an eigenvalue of the endomorphism f : E E The eigenspace of the eigenvalue λ is the set Properties E λ = {u E : f (u) λu = 0 E } E λ is a vector subspace of E 1 dim(e λ ) m λ The dimension of E λ is called the geometric multiplicity of λ Let f : E E be an endomorphism of a vector space E of dimension n The endomorphism f is diagonalizable if and only if it has n eigenvalues (counting multiplicities) and for each eigenvalue the algebraic and the geometric multiplicity coincide Corollary If f has n different eigenvalues, then it is diagonalizable Diagonalization algorithm To decide whether the endomorphism f : E E is diagonalizable or not, we will proceed as follows: (1) We find the matrix associated to f in any basis and we calculate the characteristic polynomial p f (x) (2) We find the eigenvalues and their multiplicities solving p f (x) = 0 (3) If the multiplicities of the eigenvalues add up to less than dim(e), the endomorphism does not diagonalize Otherwise go to (4) (4) For each eigenvalue λ, we find the eigenspace E λ and its dimension dim(e λ ) (5) If for all λ we have m λ = dim(e λ ), the endomorphism diagonalizes Otherwise it does not If the endomorphism diagonalizes, a basis in which it diagonalizes is given by the union of the bases of the spaces E λ 73
Math Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
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 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 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 informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
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 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 informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationDigital Workbook for GRA 6035 Mathematics
Eivind Eriksen Digital Workbook for GRA 6035 Mathematics November 10, 2014 BI Norwegian Business School Contents Part I Lectures in GRA6035 Mathematics 1 Linear Systems and Gaussian Elimination........................
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 informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationChapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
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 informationLinear Systems and Matrices
Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......
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 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 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 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 informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationMATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics
MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones
More information(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).
.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)
More informationTopic 1: Matrix diagonalization
Topic : Matrix diagonalization Review of Matrices and Determinants Definition A matrix is a rectangular array of real numbers a a a m a A = a a m a n a n a nm The matrix is said to be of order n m if it
More informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
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 information1 Last time: determinants
1 Last time: determinants Let n be a positive integer If A is an n n matrix, then its determinant is the number det A = Π(X, A)( 1) inv(x) X S n where S n is the set of n n permutation matrices Π(X, A)
More informationVectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes.
Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes Matrices and linear equations A matrix is an m-by-n array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a
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 informationCalculating determinants for larger matrices
Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det
More informationSolving a system by back-substitution, checking consistency of a system (no rows of the form
MATH 520 LEARNING OBJECTIVES SPRING 2017 BROWN UNIVERSITY SAMUEL S. WATSON Week 1 (23 Jan through 27 Jan) Definition of a system of linear equations, definition of a solution of a linear system, elementary
More informationLinear Algebra: Lecture notes from Kolman and Hill 9th edition.
Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More information1300 Linear Algebra and Vector Geometry
1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Matrix Inversion Algorithm One payoff from this theorem: It gives us a way to invert matrices.
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 207/8 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationand let s calculate the image of some vectors under the transformation T.
Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =
More informationJim Lambers MAT 610 Summer Session Lecture 1 Notes
Jim Lambers MAT 60 Summer Session 2009-0 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra
More informationMH1200 Final 2014/2015
MH200 Final 204/205 November 22, 204 QUESTION. (20 marks) Let where a R. A = 2 3 4, B = 2 3 4, 3 6 a 3 6 0. For what values of a is A singular? 2. What is the minimum value of the rank of A over all a
More informationMath Camp Notes: Linear Algebra I
Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationNOTES FOR LINEAR ALGEBRA 133
NOTES FOR LINEAR ALGEBRA 33 William J Anderson McGill University These are not official notes for Math 33 identical to the notes projected in class They are intended for Anderson s section 4, and are 2
More informationdet(ka) = k n det A.
Properties of determinants Theorem. If A is n n, then for any k, det(ka) = k n det A. Multiplying one row of A by k multiplies the determinant by k. But ka has every row multiplied by k, so the determinant
More informationNotes on Linear Algebra
1 Notes on Linear Algebra Jean Walrand August 2005 I INTRODUCTION Linear Algebra is the theory of linear transformations Applications abound in estimation control and Markov chains You should be familiar
More informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationMATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.
MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.
More informationMatrices and systems of linear equations
Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.
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 informationMATH 2030: EIGENVALUES AND EIGENVECTORS
MATH 2030: EIGENVALUES AND EIGENVECTORS Determinants Although we are introducing determinants in the context of matrices, the theory of determinants predates matrices by at least two hundred years Their
More informationChapter 2:Determinants. Section 2.1: Determinants by cofactor expansion
Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More informationQuestion: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI?
Section 5. The Characteristic Polynomial Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Property The eigenvalues
More informationLinear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02)
Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 206, v 202) Contents 2 Matrices and Systems of Linear Equations 2 Systems of Linear Equations 2 Elimination, Matrix Formulation
More information1 Matrices and Systems of Linear Equations
Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 207, v 260) Contents Matrices and Systems of Linear Equations Systems of Linear Equations Elimination, Matrix Formulation
More informationLINEAR ALGEBRA WITH APPLICATIONS
SEVENTH EDITION LINEAR ALGEBRA WITH APPLICATIONS Instructor s Solutions Manual Steven J. Leon PREFACE This solutions manual is designed to accompany the seventh edition of Linear Algebra with Applications
More informationReview for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.
Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for
More informationREVIEW FOR EXAM II. The exam covers sections , the part of 3.7 on Markov chains, and
REVIEW FOR EXAM II The exam covers sections 3.4 3.6, the part of 3.7 on Markov chains, and 4.1 4.3. 1. The LU factorization: An n n matrix A has an LU factorization if A = LU, where L is lower triangular
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 Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationChapter 2: Matrices and Linear Systems
Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers
More information1 Matrices and Systems of Linear Equations. a 1n a 2n
March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real
More informationMATH 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 informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationMatrices and Linear Algebra
Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More information2 b 3 b 4. c c 2 c 3 c 4
OHSx XM511 Linear Algebra: Multiple Choice Questions for Chapter 4 a a 2 a 3 a 4 b b 1. What is the determinant of 2 b 3 b 4 c c 2 c 3 c 4? d d 2 d 3 d 4 (a) abcd (b) abcd(a b)(b c)(c d)(d a) (c) abcd(a
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationIntroduction to Matrices and Linear Systems Ch. 3
Introduction to Matrices and Linear Systems Ch. 3 Doreen De Leon Department of Mathematics, California State University, Fresno June, 5 Basic Matrix Concepts and Operations Section 3.4. Basic Matrix Concepts
More informationChapter 1: Systems of Linear Equations
Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where
More informationMatrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices
Graphics 2009/2010, period 1 Lecture 4 Matrices m n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in
More informationAPPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF
ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done
More information1 Matrices and Systems of Linear Equations
March 3, 203 6-6. Systems of Linear Equations Matrices and Systems of Linear Equations An m n matrix is an array A = a ij of the form a a n a 2 a 2n... a m a mn where each a ij is a real or complex number.
More informationThe definition of a vector space (V, +, )
The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element
More informationFinite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.
Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form
More informationMATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.
MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of
More informationMath 1553, Introduction to Linear Algebra
Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level
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 informationFirst we introduce the sets that are going to serve as the generalizations of the scalars.
Contents 1 Fields...................................... 2 2 Vector spaces.................................. 4 3 Matrices..................................... 7 4 Linear systems and matrices..........................
More informationMath 3108: Linear Algebra
Math 3108: Linear Algebra Instructor: Jason Murphy Department of Mathematics and Statistics Missouri University of Science and Technology 1 / 323 Contents. Chapter 1. Slides 3 70 Chapter 2. Slides 71 118
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationPOLI270 - Linear Algebra
POLI7 - Linear Algebra Septemer 8th Basics a x + a x +... + a n x n b () is the linear form where a, b are parameters and x n are variables. For a given equation such as x +x you only need a variable and
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)
More informationChapter 3. Vector spaces
Chapter 3. Vector spaces Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/22 Linear combinations Suppose that v 1,v 2,...,v n and v are vectors in R m. Definition 3.1 Linear combination We say
More information1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)
1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix
More informationDaily Update. Math 290: Elementary Linear Algebra Fall 2018
Daily Update Math 90: Elementary Linear Algebra Fall 08 Lecture 7: Tuesday, December 4 After reviewing the definitions of a linear transformation, and the kernel and range of a linear transformation, we
More informationII. Determinant Functions
Supplemental Materials for EE203001 Students II Determinant Functions Chung-Chin Lu Department of Electrical Engineering National Tsing Hua University May 22, 2003 1 Three Axioms for a Determinant Function
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
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 informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationMATH 106 LINEAR ALGEBRA LECTURE NOTES
MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL - These Lecture Notes are not in a final form being still subject of improvement Contents Systems of linear equations and matrices 5 Introduction to systems of
More information3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions
A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence
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 informationMath Final December 2006 C. Robinson
Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 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 informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More information1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?
. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationElementary Linear Algebra
Matrices J MUSCAT Elementary Linear Algebra Matrices Definition Dr J Muscat 2002 A matrix is a rectangular array of numbers, arranged in rows and columns a a 2 a 3 a n a 2 a 22 a 23 a 2n A = a m a mn We
More informationMath Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT
Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear
More informationa11 a A = : a 21 a 22
Matrices The study of linear systems is facilitated by introducing matrices. Matrix theory provides a convenient language and notation to express many of the ideas concisely, and complicated formulas are
More information