(a) If U and V are vector subspaces of Rn, then U ∪ V is also a vector subspace of Rn. (b) If V and W are vector spaces of the same dimension, n, and T : V → W is any linear transformation, then it is always possible to choose bases α, of V, and β, of W, such that the matrix of T with respect to to these bases is the n×n identity matrix. (c) Let u = (2, 0, −1), v = (3, 1, 0), and w = (c, −2, 2), where c ∈ R. The set {u,v,w} is a basis of R3 only if c ≠ −7.

(a) If U and V are vector subspaces of Rn, then U ∪ V is also a vector subspace of Rn. (b) If V and W are vector spaces of the same dimension, n, and T : V → W is any linear transformation, then it is always possible to choose bases α, of V, and β, of W, such that the matrix of T with respect to to these bases is the n×n identity matrix. (c) Let u = (2, 0, −1), v = (3, 1, 0), and w = (c, −2, 2), where c ∈ R. The set {u,v,w} is a basis of R3 only if c ≠ −7.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.6: The Matrix Of A Linear Transformation

Problem 43EQ

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