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6. Let V be a vector space of dimension n and W a vector space of dimension m. Let L: V → W be a linear transformation such that the dimension of the kernel of L is r. Show that there are ordered bases E = U1, ..., ĩn] of V and F = [W1, ..., wm] of W such that the matrix representation of L with respect to E and F is an m x n matrix A = =(a;) satisfying aij = 1 if i = j and i

6. Let V be a vector space of dimension n and W a vector space of dimension m. Let L: V → W be a linear transformation such that the dimension of the kernel of L is r. Show that there are ordered bases E = U1, ..., ĩn] of V and F = [W1, ..., wm] of W such that the matrix representation of L with respect to E and F is an m x n matrix A = =(a;) satisfying aij = 1 if i = j and i

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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Let V be a vector space of dimension n and W a vector space of → W be a linear transformation such that the dimension = [01,..., Un] of V and F = w1, ... , wm of W such that the matrix representation of L with respect to E and 6. dimension m. Let L: V of the kernel of L is r. Show that there are ordered bases E F is an m x n matrix A = (aij) satisfying = 1 if i = j and i <n – r Aij Aij = 0 otherwise
Transcribed Image Text:Let V be a vector space of dimension n and W a vector space of → W be a linear transformation such that the dimension = [01,..., Un] of V and F = w1, ... , wm of W such that the matrix representation of L with respect to E and 6. dimension m. Let L: V of the kernel of L is r. Show that there are ordered bases E F is an m x n matrix A = (aij) satisfying = 1 if i = j and i <n – r Aij Aij = 0 otherwise
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