Let V and W be vector spaces over a field K having bases α = {V₁, V₂} and B = {w₁, W2, W3} respectively. Recall that the set of linear transformations (V, W) = {T: V → W | T is linear} is a vector space. For i = 1,2 and j = 1,2,3 define Tij : V → W by if k = i, if k i Recall that by Theorem 34, the above information is sufficient to define each Tij as a linear transformation since it specifies the value on the basis {V₁, V₂}. (a) Let S = 2T12 - 3T23 + 4T₁1. Evaluate S(2v₁ - 30₂) and compute the matrix [S]. (b) (c) ] Prove that A = {T11, T12, T13, T21, T22, T23} is a basis of (V, W). 0 (36) 0 1 Let B = Tij (Uk) = { W ₁ 0 :(V, W). → M 3x2 (K) by 0 be the standard basis of M 3x2 (K). Define $(T) = [T] and recall from Theorem 46 that is an isomorphism (in particular, is a linear transformation). Compute its matrix [$13.

Let V and W be vector spaces over a field K having bases α = {V₁, V₂} and B = {w₁, W2, W3} respectively. Recall that the set of linear transformations (V, W) = {T: V → W | T is linear} is a vector space. For i = 1,2 and j = 1,2,3 define Tij : V → W by if k = i, if k i Recall that by Theorem 34, the above information is sufficient to define each Tij as a linear transformation since it specifies the value on the basis {V₁, V₂}. (a) Let S = 2T12 - 3T23 + 4T₁1. Evaluate S(2v₁ - 30₂) and compute the matrix [S]. (b) (c) ] Prove that A = {T11, T12, T13, T21, T22, T23} is a basis of (V, W). 0 (36) 0 1 Let B = Tij (Uk) = { W ₁ 0 :(V, W). → M 3x2 (K) by 0 be the standard basis of M 3x2 (K). Define $(T) = [T] and recall from Theorem 46 that is an isomorphism (in particular, is a linear transformation). Compute its matrix [$13.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 38EQ

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