3 Matrices - Solutions

3. Matrices (References: Comps Study Guide for Linear Algebra Section 3; Course in Linear Algebra, Chapters 2 and 3) Coordinate Vectors: Let V be a vector space and ↵ = {~v1 , . . . , ~vn } be a basis for V . Then for any ~v 2 V , there are unique coefficients a1 , . . . , an 2 R such that ~v = a1~v1 + · · · + an~vn . In this notation, the coordinate vector of ~v with respect to ↵ is [~v ]↵ = Matrix of a Linear Transformation: Let T : V ! W be a linear transformation and let ↵ = {~v1 , . . . , ~vn } and = {w ~ 1, . . . , w ~ n } be bases for V and W , respectively. Then the matrix of T with respect to ↵ and , denoted [T ]↵ , is the matrix whose ith column is [T (~vi )] , the coordinate vector of T (~vi ) with respect to . Ex. ⇢✓ 1. Let M be ✓ the vector ◆ 2⇥2 ✓ (R) ◆ ◆ ✓space◆of 2 ⇥ 2 matrices with real coefficients and let 1 0 0 1 0 0 0 0 ↵= , , , be its standard basis. Let T : M2⇥2 (R) ! M2⇥2 (R) by T (A) = 0 0 0 0 1 0 0 1 At where At is the transpose of the matrix A. Compute the matrix of T with respect to the basis ↵. Theorem: Let T : V ! W be a linear transformation between finite-dimensional vector spaces V and W . If A = [T ]↵ where ↵ and are any bases of V and W , respectively, then (1) T is one-to-one if and only if nullity(A) = 0. (2) T is onto if and only if the rank(A) = dim(W ). Ex. 2. Let T : M2⇥2 (R) ! M2⇥2 (R) be the transpose map from Ex. 1. Is T one-to one? Is T onto? Using the Matrix of T to find T (~v ): Let T : V ! W and ↵ and be bases of the vector spaces V and W , respectively. For a vector ~v 2 V , write down the equation that relates the coordinate vector of T (~v ) to the coordinate vector of ~v and the matrix of T . Matrix of a Composition: Let T : V ! W and S : W ! X be linear transformations and let ↵, , be bases of the vector spaces V, W , and X, respectively. Write down the equation that relates the matrix of the composition ST to the matrices of S and T . Ex. 3. Let V = R4 and W = P2 (R) be the vector space of polynomials with real coefficients and degree at most 2. Let ↵ = {~e1 , ~e2 , ~e3 , ~e4 } be the standard basis of R4 and let = {1, x + 1, x2 + x + 1}. It is a fact, which you may assume, that is a basis for W . Suppose that T :3V ! W is a linear transformation and 2 1 2 3 4 15. Find T (0, 2, 1, 3). the matrix of T with respect to ↵ and is [T ]↵ = 42 0 0 0 2 3 1 Invertible Matrices: Explain what it means for an n ⇥ n matrix A to be invertible. (Note: Only square matrices can be invertible.) Theorem: Let A be an n ⇥ n matrix. The following are equivalent: (1) A is invertible. (2) The columns of A are linearly independent. (3) The rows of A span Rn . (4) The columnspace (i.e., range or image) of A is Rn . (5) The nullspace (i.e., kernel) of A is {0}. (6) rank(A) = n. (7) nullity(A) = 0. (8) det(A) 6= 0. (9) = 0 is not an eigenvalue of A. 0 1 1 1 2 Ex. 4. Consider the matrix A = @0 0 1A. 1 2 4 (a) Compute det(A). (b) Is A invertible? If so, compute the inverse of A. Invertible Maps: Let T : V ! W be a linear transformation. Explain what it means for T to be invertible. Theorem: A map T is invertible if and only if T is one-to-one and onto. Theorem: Let T : V ! W be a linear transformation between finite-dimensional vector spaces. If A = [T ]↵ where ↵ and are any bases of V and W , respectively, then T is invertible if and only if A is invertible. Ex. 5. Let T : V ! V be a linear transformation and let v1 , ~v2 } be a basis for the vector space V . ✓ ↵ = {~◆ 3 1 ↵ Suppose that the matrix of T with respect to ↵ is [T ]↵ = . 2 1 (a) Explain how you know that T is invertible. (b) Calculate T 1 (~v1 ). Write your answer as a linear combination of the vectors ~v1 and ~v2 . Change of Basis: Recall that the identity map I : V ! V satisfies I(~v ) = ~v for all ~v 2 V . Let ↵ and be bases of V . In this case, the matrix [I]↵ is called the change of basis matrix from ↵ to . both Inverse of a Change of Basis Matrix: Since the identity map is invertible, so is any change of basis matrix. What is [I]↵ 1 ? Changing Coordinates for the Matrix of a Transformation: Suppose that T : V ! W , ↵ and ↵0 are bases for V and and 0 0 0 are bases for W . Given [T ]↵ , write down expressions for [T ]↵ , [T ]↵0 , and [T ]↵0 . ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ 1 8 1 2 = . = and T 1 1 1 3 (a) Find the matrix of T with respect to the standard basis of R2 . (b) Is T one-to-one? Is T onto? Justify your answers. Ex. 6. Let T : R2 ! R2 be a linear transformation such that T Additional Problems Ex. 7. Let Pn (R) be the vector R x space of polynomials with real coefficients of degree at most n. Define T : P2 (R) ! P3 (R) by T (f ) = 0 f (t) dt. (a) Compute the matrix of T with respect to the standard bases {1, x, x2 } of P2 (R) and {1, x, x2 , x3 } of P3 (R). (b) Is T one-to one? Is T onto? (c) Find bases for ker(T ) and Im(T ). Ex. ⇢✓ 8. Let M be ✓ the vector ◆ 2⇥2 ✓ (R) ◆ ◆ ✓space◆of 2 ⇥ 2 matrices with real coefficients and let 1 0 0 1 0 0 0 0 ↵= , , , be its standard basis. 0 0 0 0 1 0 0 1 ✓ ◆ a b (a) Let T : M2⇥2 (R) ! R by T (A) = a + b + c + d where A = . Compute the matrix c d respect to the bases ↵ for M2⇥2 (R) and = {1} for R. (b) Find a basis for ker(T ). (c) Is T one-to one? Is T onto? 2 1 1 4 3 3 Ex. 9. Let T : R5 ! R3 be the linear transformation with matrix representation [T ]std = std 5 5 5 3 with respect to the standard bases on R and R . Find a basis for Ker(T ) and Im(T ). Ex. 10. Consider the linear transformation T : R3 ! R3 defined by T (x, y, z) = (2x + y Determine whether or not T is invertible, and if so, find a formula for T 1 (x, y, z). 4z, 4y Ex. 11. Let A and B be invertible n ⇥ n matrices. Show that AB is invertible and (AB) (Note: This is a common result that you could usually use without proof.) Ex. 12. Let ↵ = {~v1 , ~v2 , ~v3 } and 1 of T with 1 3 7 3 1 35 7 1 3 7 5z, z). =B 1 A 1 . = {w ~ 1, w ~ 2, w ~ 3 } be bases for a vector space V . Suppose that ~v1 = w ~1 ~v2 = Compute the change of basis matrix from 2w ~2 w ~2 w ~3 ~v3 = 2w ~2 w ~ 3. 2w ~3 to ↵. Ex. 13. Let P2 be the vector space of polynomials with real coefficients of degree at most 2, and let ↵ = {1, x + 1, x2 + x + 1}. It is a fact, which you may assume, that ↵ is a basis for 2 V . Suppose 3 that 1 2 3 43 0 15. Find T : V ! V is a linear transformation and the matrix of T relative to ↵ is [T ]↵ ↵ = 2 5 1 T (3x2 + x + 2).  p(1) Ex. 14. Let P2 = {a + bt + ct2 : a, b, c 2 R} and T : P2 ! R2 be defined by T (p) = 0 . You may p (1) assume that T is linear. ⇢  1 1 2 (a) Find the matrix representation of T with respect to the bases {1, t, t } and , . 0 1 (b) Find the rank and nullity of T . (c) Find bases of the kernel and image of T .