University:
Amherst College
Course:
MATH 271 | Linear Algebra
Academic year:

2022
Views:

388

Pages:

1
Author:

pigamdomo81d

Linear Algebra: Review Session 3 Matrices Matrix of a Linear Transformation and Coordinate Vectors 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. 0 0 0 0 1 0 0 1 (a) Let T : M2×2 (R) → M2×2 (R) by T (A) = At where At is the transpose of the matrix A. Compute the matrix of T with respect to the basis α. Is T one-to one? Is Tonto? a b (b) Let T : M2×2 (R) → R by T (A) = a + b + c + d where A = . Compute the matrix of T with c d respect to the bases α for M2×2 (R) and β = {1} for R. Is T one-to one? Is T onto? Ex. 2. Let Pn be theR vector space of polynomials with real coefficients of degree at most n. Define x T : P2 → P3 by T (f ) = 0 f (t) dt. (a) Prove that T is linear. (b) Compute T (ax2 + bx + c) where a, b, c ∈ R. (c) Compute the matrix of T with respect to the standard bases {1, x, x2 } of P2 and {1, x, x2 , x3 } of P3 .   1 1 1 1 1   Ex. 3. Let T : R5 → R3 be the linear transformation with matrix representation [T ]std std = 3 3 3 3 3 5 5 7 7 7 with respect to the standard bases on R5 and R3 . Find a basis for Ker(T ) and Im(T ). Invertible Matrices and Invertible Maps  1 Ex. 4. Compute the inverse of the matrix A = 0 1 1 0 2  2 1. 4 Ex. 5. 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. 6. (a) Let U and W be vector spaces. Explain what it means to say that a linear transformation S : U → W is invertible. (b) Let T : V → V be a linear transformation and let α= {v1 , v2 } be a basis for the vector space V . Suppose 3 −1 that the matrix of T with respect to α is . Explain how you know that T is invertible. 2 1 (c) Calculate T −1 (v1 ). Write your answer as a linear combination of the vectors v1 and v2 . Change of Basis 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. 7. Let T : R2 → R2 be a linear transformation such that T Ex. 8. 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  V . Suppose  that 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 . You may assume Ex. 9. Let P2 = {a + bt + ct : a, b, c ∈ R} and T : P2 → R be defined by T (p) = 0 p (1) 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 .

Linear Algebra Review Session 3 Matrices

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