MATH 423 Linear Algebra II Lecture 10: Inverse matrix. Change of coordinates.

Transcription

1 MATH 423 Linear Algebra II Lecture 10: Inverse matrix. Change of coordinates.

2 Let V be a vector space and α = [v 1,...,v n ] be an ordered basis for V. Theorem 1 The coordinate mapping C : V F n given by C(v) = [v] α is linear and invertible (i.e., one-to-one and onto). Let W be another vector space and β = [w 1,...,w m ] be an ordered basis for W. Theorem 2 The mapping M : L(V,W) M m,n (F) given by M(L) = [L] β α is linear and invertible.

3 Linear maps and matrix multiplication Let V, W, and X be vector spaces. Suppose α = [v 1,...,v n ] is an ordered basis for V, β = [w 1,...,w m ] is an ordered basis for W, and γ = [x 1,...,x k ] is an ordered basis for X. Theorem 1 For any linear transformation L : V W and any vector v V, [L(v)] β = [L] β α[v] α. Theorem 2 For any linear transformations L : V W and T : W X, [T L] γ α = [T] γ β [L]β α. Theorem 3 For any linear operators L : V V and T : V V, [T L] α = [T] α [L] α.

4 Identity matrix Definition. The identity matrix (or unit matrix) is an n n matrix I = (a ij ) such that a ii = 1 and a ij = 0 for i j. It is also denoted I n. I 1 = (1), I 2 = ( ) 1 0, I = In general, I = Theorem. Let A be an arbitrary m n matrix. Then I m A = AI n = A.

5 Inverse matrix Definition. Let A M n,n (F). Suppose there exists an n n matrix B such that AB = BA = I n. Then the matrix A is called invertible and B is called the inverse of A (denoted A 1 ). AA 1 = A 1 A = I Basic properties of inverse matrices: If B = A 1 then A = B 1. In other words, if A is invertible, so is A 1, and A = (A 1 ) 1. The inverse matrix (if it exists) is unique. If n n matrices A and B are invertible, so is AB, and (AB) 1 = B 1 A 1. Similarly, (A 1 A 2...A k ) 1 = A 1 k...a 1 2 A 1 1.

6 Inverting 2 2 matrices Definition. ( ) The determinant of a 2 2 matrix a b A = is det A = ad bc. c d ( ) a b Theorem A matrix A = is invertible if c d and only if det A 0. If det A 0 then ( ) 1 a b = c d 1 ad bc ( ) d b. c a

7 ( ) a b Theorem A matrix A = is invertible if c d and only if det A 0. If det A 0 then ( ) 1 ( ) a b 1 d b =. c d ad bc c a ( d b Proof: Let B = c a AB = BA = ). Then ( ad bc 0 0 ad bc ) = (ad bc)i 2. In the case det A 0, we have A 1 = (det A) 1 B. In the case det A = 0, the matrix A is not invertible as otherwise AB = O = A 1 (AB) = A 1 O = O = (A 1 A)B = O = I 2 B = O = B = O = A = O, but the zero matrix is not invertible.

8 Left multiplication Any m n matrix A M m,n (F) gives rise to a linear transformation L A : F n F m given by L A (x) = Ax, where x F n and L(x) F m are regarded as column vectors. Theorem 1 The matrix of the transformation L A relative to the standard bases in F n and F m is exactly A. Theorem 2 Suppose L : F n F m is a linear map. Then there exists an m n matrix A such that L(x) = Ax for all x F n. Columns of A are vectors L(e 1 ), L(e 2 ),...,L(e n ), where e 1,e 2,...,e n is the standard basis for F n.

9 Matrix of a linear transformation (revisited) Let V,W be vector spaces and f : V W be a linear map. Let α = [v 1,v 2,...,v n ] be a basis for V and g 1 : V F n be the coordinate mapping corresponding to this basis. Let β = [w 1,...,w m ] be a basis for W and g 2 : W F m be the coordinate mapping corresponding to this basis. V g 1 f W g2 F n F m The composition g 2 f g 1 1 is a linear mapping of F n to F m. It is uniquely represented as x Ax, where A M m,n (F). Theorem A = [f ] β α, the matrix of the transformation f relative to the bases α and β.

10 Change of coordinates Let V be a vector space of dimension n. Let v 1,v 2,...,v n be a basis for V and g 1 : V F n be the coordinate mapping corresponding to this basis. Let u 1,u 2,...,u n be another basis for V and g 2 : V F n be the coordinate mapping corresponding to this basis. V g 1 g 2 ւ ց F n F n The composition g 2 g 1 1 is a linear operator on F n. It has the form x Ux, where U is an n n matrix. U is called the transition matrix from v 1,v 2...,v n to u 1,u 2...,u n. Columns of U are coordinates of the vectors v 1,v 2,...,v n with respect to the basis u 1,u 2,...,u n.

11 Problem. Find the transition matrix from the basis v 1 = (1, 2, 3), v 2 = (1, 0, 1), v 3 = (1, 2, 1) to the basis u 1 = (1, 1, 0), u 2 = (0, 1, 1), u 3 = (1, 1, 1). It is convenient to make a two-step transition: first from v 1,v 2,v 3 to e 1,e 2,e 3, and then from e 1,e 2,e 3 to u 1,u 2,u 3. Let U 1 be the transition matrix from v 1,v 2,v 3 to e 1,e 2,e 3 and U 2 be the transition matrix from u 1,u 2,u 3 to e 1,e 2,e 3 : U 1 = 2 0 2, U 2 =

12 Basis v 1,v 2,v 3 = coordinates x Basis e 1,e 2,e 3 = coordinates U 1 x Basis u 1,u 2,u 3 = coordinates U 1 2 (U 1x)=(U 1 2 U 1)x Thus the transition matrix from v 1,v 2,v 3 to u 1,u 2,u 3 is U2 1 U U2 1 U 1 = = =

13 Problem. Consider a linear operator L : F 2 F 2, ( ) ( ) ( ) x 1 1 x L =. y 0 1 y Find the matrix of L with respect to the basis v 1 = (3, 1), v 2 = (2, 1). Let N be the desired matrix. Columns of N are coordinates of the vectors L(v 1 ) and L(v 2 ) w.r.t. the basis v 1,v 2. ( ) ( ( ) ( L(v 1 ) = =, L(v 0 1)( 1 1) 2 ) = =. 0 1)( 1 1) Clearly, L(v 2 ) = v 1 = 1v 1 + 0v 2. L(v 1 ) = av 1 + bv 2 ( ) 2 1 Thus N =. 1 0 { 3a + 2b = 4 a + b = 1 { a = 2 b = 1

14 Change of coordinates for a linear operator Let L : V V be a linear operator on a vector space V. Let A be the matrix of L relative to a basis a 1,a 2,...,a n for V. Let B be the matrix of L relative to another basis b 1,b 2,...,b n for V. Let U be the transition matrix from the basis a 1,a 2,...,a n to b 1,b 2,...,b n. a-coordinates of v U b-coordinates of v A B a-coordinates of L(v) U b-coordinates of L(v) It follows that UAx = BUx for all x F n = UA = BU. Then A = U 1 BU and B = UAU 1.

15 Problem. Consider a linear operator L : F 2 F 2, ( ) ( ) ( ) x 1 1 x L =. y 0 1 y Find the matrix of L with respect to the basis v 1 = (3, 1), v 2 = (2, 1). Let S be the matrix of L with respect to the standard basis, N be the matrix of L with respect to the basis v 1,v 2, and U be the transition matrix from v 1,v 2 to e 1,e 2. Then N = U 1 SU. ( ) ( ) S =, U =, ( ) ( ) ( ) N = U 1 SU = ( ) ( ) ( ) = =