babeeb0oL

2020-11-30

Assum T:

Linearity Properties:

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

casincal

Skilled2020-12-01Added 82 answers

Proof:

Let T be defined as$T(v)=Av$

Take$v=\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\\ .\\ .\\ {v}_{m}\end{array}\right]\text{}\in \text{}{R}_{m}\text{}and\text{}A=\left[\begin{array}{ccccc}{a}_{11}& {a}_{12}& .& .& {a}_{1m}\\ {a}_{21}& {a}_{22}& .& .& {a}_{2m}\\ .& .& .& .& .\\ .& .& .& .& .\\ {a}_{n1}& {a}_{n2}& .& .& {a}_{nm}\end{array}\right].$

Further obtain the result as follows:

$T(v)=Av$

$=\left[\begin{array}{ccccc}{a}_{11}& {a}_{12}& .& .& {a}_{1m}\\ {a}_{21}& {a}_{22}& .& .& {a}_{2m}\\ .& .& .& .& .\\ .& .& .& .& .\\ {a}_{n1}& {a}_{n2}& .& .& {a}_{nm}\end{array}\right]\text{}\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\\ .\\ .\\ {v}_{m}\end{array}\right]$

$\left[\begin{array}{ccccc}{a}_{11}{v}_{1}& +& {a}_{12}{v}_{2}& +\cdots \text{}+& {a}_{1m}{v}_{m}\\ {a}_{21}{v}_{1}& +& {a}_{22}{v}_{2}& +\cdots \text{}+& {a}_{2m}{v}_{m}\\ .& .& .& .& .\\ .& .& .& .& .\\ {a}_{n1}{v}_{1}& +& {a}_{n2}{v}_{2}& +\cdots \text{}+& {a}_{nm}{v}_{m}\end{array}\right]$

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.

Let T be defined as

Take

Further obtain the result as follows:

As the colums of A are linearly idependent, no rows Av will be equal. Thus, the vector obtained is different than one another in

Hence, if the columns of A are linearly independent, then T is one to one.

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