 # Assum T: R^m to R^n is a matrix transformation with matrix A.Prove that if the column babeeb0oL 2020-11-30 Answered

Assum 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
$A0=0$
$A\left(cv\right)=cAv$

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Proof:
Let T be defined as $T\left(v\right)=Av$
Take
Further obtain the result as follows:
$T\left(v\right)=Av$

As the colums of A are linearly idependent, no rows Av will be equal. Thus, the vector obtained is different than one another in ${R}_{n}.$
Hence, if the columns of A are linearly independent, then T is one to one.