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Found in: Page 53

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Question 7: Suppose $\stackrel{\mathbf{\to }}{{\mathbf{v}}_{\mathbf{1}}}{\mathbf{,}}\stackrel{\mathbf{\to }}{{\mathbf{v}}_{\mathbf{2}}}{\mathbf{,}}\stackrel{\mathbf{\to }}{{\mathbf{v}}_{\mathbf{3}}}{\mathbf{\cdots }}\stackrel{\mathbf{\to }}{{\mathbf{v}}_{\mathbf{m}}}$are arbitrary vectors in Consider the linear transformation from to given by ${\mathbit{T}}\left[{x}_{1}\phantom{\rule{0ex}{0ex}}{x}_{2}\phantom{\rule{0ex}{0ex}}⋮\phantom{\rule{0ex}{0ex}}{x}_{m}\right]{\mathbf{}}{\mathbf{=}}\left[{x}_{1}\stackrel{\to }{{v}_{1}},{x}_{2}\stackrel{\to }{{v}_{2}},{x}_{3}\stackrel{\to }{{v}_{3}}\cdots {x}_{m}\stackrel{\to }{{v}_{m}}\right]$

Yes, given transformation is linear and the matrix represented by $A=\left[\stackrel{\to }{{v}_{1}},\stackrel{\to }{{v}_{2}},\stackrel{\to }{{v}_{3}}\cdots \stackrel{\to }{{v}_{m}}\right]$.

See the step by step solution

## Step1: System of transformation

We have given a transformation with$T\left[{\mathrm{x}}_{1}\phantom{\rule{0ex}{0ex}}{\mathrm{x}}_{2}\phantom{\rule{0ex}{0ex}}⋮\phantom{\rule{0ex}{0ex}}{\mathrm{x}}_{\mathrm{m}}\right]=\left[{\mathrm{x}}_{1}\stackrel{\to }{{\mathrm{v}}_{1}},{\mathrm{x}}_{2}\stackrel{\to }{{\mathrm{v}}_{2}},{\mathrm{x}}_{3}\stackrel{\to }{{\mathrm{v}}_{3}}\cdots {\mathrm{x}}_{\mathrm{m}}\stackrel{\to }{{\mathrm{v}}_{\mathrm{m}}}\right]\phantom{\rule{0ex}{0ex}}$ .

## Step2: Linear Transformation

A transformation from ${R}^{m}\to {R}^{n}$ is said to be linear if the following condition holds:

1. Identity of Rm should be mapped to identity of Rn.
2. $T\left(a+b\right)=T\left(a\right)+T\left(b\right)$For all$a,b\in {R}^{m}$ .
3. $T\left(ca\right)=cT\left(a\right)$Where c is any scalar and $a\in {R}^{m}$.

## Step3: Checking for linear transformation

Now we will find the transformation of identity element.

$T\left(0,0...0\right)=\left(0,0...0\right)$

Now let $a,b\in {R}^{m}$.

Then the transformation

$T\left(\alpha a+\beta b\right)=\alpha T\left(a\right)+\beta T\left(b\right)$

Thus, all the conditions are true.

## Step4: Matrix for linear transformation

The matrix represented by.$A=\left[\stackrel{\to }{{v}_{1}},\stackrel{\to }{{v}_{2}},\stackrel{\to }{{v}_{3}}\cdots \stackrel{\to }{{v}_{m}}\right]$

Hence, yes the given transformation is linear and the matrix represented by $A=\left[\stackrel{\to }{{v}_{1}},\stackrel{\to }{{v}_{2}},\stackrel{\to }{{v}_{3}}\cdots \stackrel{\to }{{v}_{m}}\right]$.