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Expert-verified Found in: Page 55 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Consider a linear transformation T from ${{\mathbit{R}}}^{{\mathbf{2}}}$ to ${{\mathbit{R}}}^{{\mathbf{2}}}$. Suppose that $\stackrel{\mathbf{\to }}{\mathbf{v}}$ and $\stackrel{\mathbf{\to }}{\mathbf{w}}$ are two arbitrary vectors in ${{\mathbit{R}}}^{{\mathbf{2}}}$ and that $\stackrel{\mathbf{\to }}{\mathbf{x}}$ is a third vector whose endpoint is on the line segment connecting the endpoints of $\stackrel{\mathbf{\to }}{\mathbf{v}}$ and $\stackrel{\mathbf{\to }}{\mathbit{w}}$. Is the endpoint of the vector${\mathbit{T}}{\mathbf{\left(}}\stackrel{\mathbf{\to }}{\mathbf{x}\mathbf{\right)}}$ necessarily on the line segment connecting the endpoints of ${\mathbit{T}}{\mathbf{\left(}}\stackrel{\mathbf{\to }}{\mathbf{v}\mathbf{\right)}}$ and ${\mathbit{T}}{\mathbf{\left(}}\stackrel{\mathbf{\to }}{\mathbit{w}\mathbf{\right)}}$ ? Justify your answer. Thus, we can say that $T\left(\stackrel{\to }{x}\right)=T\left(\stackrel{\to }{v}\right)+k\left(T\left(\stackrel{\to }{w}\right)-T\stackrel{\to }{\left(v}\right)\right)$Is on line segment.

See the step by step solution

## Step by step Explanation Step1: Assuming the equation

We can write $\stackrel{\to }{x}=\stackrel{\to }{v}+k\left(\stackrel{\to }{w}-\stackrel{\to }{v}\right)$ where k is any scalar between 0 and 1.

## Step2: Linear Transformation

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

1. Identity of ${R}^{m}$ should be mapped to identity of role="math" localid="1659717675430" ${R}^{n}$.
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: Solving the equations

On a similar basis we can write

$T\left(\stackrel{\to }{v}+k\left(\stackrel{\to }{w}-\stackrel{\to }{v}\right)=T\left(\stackrel{\to }{v}\right)+kT\left(\stackrel{\to }{w}-\stackrel{\to }{v}\right)\phantom{\rule{0ex}{0ex}}T\left(\stackrel{\to }{v}+k\left(\stackrel{\to }{w}-\stackrel{\to }{v}\right)=T\left(\stackrel{\to }{v}\right)+k\left(T\left(\stackrel{\to }{w}\right)-T\stackrel{\to }{\left(v}\right)\right)$

Thus, we can say that $T\left(\stackrel{\to }{x}\right)=T\left(\stackrel{\to }{v}\right)+k\left(T\left(\stackrel{\to }{w}\right)-T\stackrel{\to }{\left(v}\right)\right)$ Is on line segment. ### Want to see more solutions like these? 