Let p, m, and n be positive integers and F a field. Let V be the space of m ×n matrices over F and W the space of p ×n matrices over F. Let B be a fixed p ×m matrix and let T be the linear transformation from V into W defined by T(A) = BA. Prove that T is invertible if and only if p = m and B is an invertible m ×m matrix.

Let p, m, and n be positive integers and F a field. Let V be the space of m ×n matrices over F and W the space of p ×n matrices over F. Let B be a fixed p ×m matrix and let T be the linear transformation from V into W defined by T(A) = BA. Prove that T is invertible if and only if p = m and B is an invertible m ×m matrix.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.5: Applications

Problem 30EQ

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