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
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d48b152-3ad9-46ac-ae5d-bdce00824bf1%2Fdafc4b16-3e25-4d33-be73-cfb98ed0edb4%2Fi857pdc_processed.jpeg&w=3840&q=75)
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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