Matrices

Matrices Coordinate Vectors: Let V be a vector space and α = {~v1 , . . . , ~vn } be a basis for V . Then for any ~v ∈ V , there are unique coefficients a1 , . . . , an ∈ R such that ~v = a1~v1 + · · · + an~vn . In this notation, the coordinate vector of ~v with respect to α is [~v ]α = Matrix of a Linear Transformation: Let T : V → W be a linear transformation and let α = {~v1 , . . . , ~vn } and β = {w ~ 1, . . . , w ~ n } be bases for V and W , respectively. Then the matrix of T with respect to α and β, denoted [T ]βα , is the matrix whose ith column is [T (~vi )]β , the coordinate vector of T (~vi ) with respect to β. Ex.  1. Let M be  the vector of 2 × 2 matrices with real coefficients and let  2×2  (R)   space 1 0 0 1 0 0 0 0 α= , , , 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 α. Theorem: Let T : V → W be a linear transformation between finite-dimensional vector spaces V and W . If A = [T ]βα where α and β 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 α and β be bases of the vector spaces V and W , respectively. For a vector ~v ∈ 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 α, β, γ 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 α = {~e1 , ~e2 , ~e3 , ~e4 } be the standard basis of R4 and let β = {1, x + 1, x2 + x + 1}. It is a fact, which you may assume, that β is a basis for W . Suppose that T :V → W is a linear transformation and  1 2 3 4 the matrix of T with respect to α and β is [T ]βα = 2 0 0 −1. Find T (0, 2, −1, 3). 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) λ = 0 is not an eigenvalue of A.  1 1 2 Ex. 4. Consider the matrix A = 0 0 1. 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 ]βα where α and β 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 .  α = {~ 3 −1 Suppose that the matrix of T with respect to α is [T ]α . α = 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 ∈ V . Let α and β both be bases of V . In this case, the matrix [I]βα is called the change of basis matrix from α to β. Inverse of a Change of Basis Matrix: Since the identity map is invertible, so is any change of basis matrix.
−1 ? What is [I]βα Changing Coordinates for the Matrix of a Transformation: Suppose that T : V → W , α and α0 are bases 0 0 for V and β and β 0 are bases for W . Given [T ]βα , write down expressions for [T ]βα , [T ]βα0 , and [T ]βα0 .         1 8 1 −2 = . Ex. 6. Let T : R → R be a linear transformation such that T = 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. 2 2 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 of 2 × 2 matrices with real coefficients and let  2×2  (R)   space 1 0 0 1 0 0 0 0 α= , , , 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 α for M2×2 (R) and β = {1} for R. (b) Find a basis for ker(T ). (c) Is T one-to one? Is T onto?  1 1  3 3 Ex. 9. Let T : R5 → R3 be the linear transformation with matrix representation [T ]std = std 5 5 with respect to the standard bases on R5 and R3 . Find a basis for Ker(T ) and Im(T ). of T with 1 3 7 1 3 7  1 3 7 Ex. 10. Consider the linear transformation T : R3 → R3 defined by T (x, y, z) = (2x + y − 4z, 4y − 5z, −z). Determine whether or not T is invertible, and if so, find a formula for T −1 (x, y, z). Ex. 11. Let A and B be invertible n × n matrices. Show that AB is invertible and (AB)−1 = B −1 A−1 . (Note: This is a common result that you could usually use without proof.) Ex. 12. Let α = {~v1 , ~v2 , ~v3 } and β = {w ~ 1, w ~ 2, w ~ 3 } be bases for a vector space V . Suppose that ~v1 = w ~ 1 − 2w ~ 2 − 2w ~3 ~v2 = −w ~2 − w ~3 ~v3 = 2w ~2 − w ~ 3. Compute the change of basis matrix from β to α. Ex. 13. Let P2 be the vector space of polynomials with real coefficients of degree at most 2, and let α = {1, x + 1, x2 + x + 1}. It is a fact, which you may assume, that α is a basis for  that  V . Suppose 1 2 3 3 0 −1. Find T : V → V is a linear transformation and the matrix of T relative to α is [T ]α α = 2 5 1 T (3x2 + x + 2).   p(1) 2 2 Ex. 14. Let P2 = {a + bt + ct : a, b, c ∈ R} and T : P2 → R 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 .