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Asssume T: R^m to R^n is a matrix transformation with matrix A. Prove that if the columns of A are l

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 44EQ

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Question

Asssume T: R^m to R^n is a matrix transformation with matrix A.

Prove that if the columns of A are linearly independent, then T is one to one (i.e injective). (Hint: Remember that matrix transformations satisfy the linearity properties.

Linearity Properties:

If A is a matrix, v and w are vectors and c is a scalar then

A0=0

A(cv)= cAv

A(v+w)=Av+Aw

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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