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Found in: Page 53

### Linear Algebra With Applications

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

# Use the concept of a linear transformation in terms of the formula ${\mathbit{y}}{\mathbf{=}}{\mathbit{A}}\stackrel{\mathbf{⇀}}{\mathbf{x}}$, and interpret simple linear transformations geometrically. Find the inverse of a linear transformation from localid="1659964769815" ${{\mathbf{ℝ}}}^{2}{\mathbf{}}{\mathbit{t}}{\mathbit{o}}{\mathbf{}}{{\mathbf{ℝ}}}^{2}{\mathbf{}}$to (if it exists). Find the matrix of a linear transformation column by column.Consider the transformations from ${{\mathbf{ℝ}}}^{{\mathbf{3}}}{\mathbf{}}{\mathbit{t}}{\mathbit{o}}{\mathbf{}}{{\mathbf{ℝ}}}^{{\mathbf{3}}}{\mathbf{}}$ defined in Exercises 1 through 3. Which of these transformations are linear?${{\mathbit{y}}}_{{\mathbf{1}}}{\mathbf{=}}{\mathbf{2}}{{\mathbit{x}}}_{{\mathbf{2}}}\phantom{\rule{0ex}{0ex}}{{\mathbit{y}}}_{{\mathbf{2}}}{\mathbf{=}}{{\mathbit{x}}}_{{\mathbf{2}}}{\mathbf{+}}{\mathbf{2}}\phantom{\rule{0ex}{0ex}}{{\mathbit{y}}}_{{\mathbf{3}}}{\mathbf{=}}{\mathbf{2}}{{\mathbit{x}}}_{{\mathbf{2}}}$

The given transformation${y}_{1}=2{x}_{2}\phantom{\rule{0ex}{0ex}}{y}_{2}={x}_{2}+2\phantom{\rule{0ex}{0ex}}{y}_{3}=2{x}_{2}$ is not linear.

See the step by step solution

## Step1: System of equations

We have given that system of equation is .${y}_{1}=2{x}_{2}\phantom{\rule{0ex}{0ex}}{y}_{2}={x}_{2}+2\phantom{\rule{0ex}{0ex}}{y}_{3}=2{x}_{2}$

For the transformation of ${R}^{3}to{R}^{3}$ it can be written as:

$T\left({x}_{1},{x}_{2},{x}_{3}\right)=\left({y}_{1},{y}_{2},{y}_{3}\right)\phantom{\rule{0ex}{0ex}}T\left({x}_{1},{x}_{2},{x}_{3}\right)=\left(2{x}_{2},{x}_{2}+2,2{x}_{2}\right)\phantom{\rule{0ex}{0ex}}$

## 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. For all $T\left(a+b\right)=T\left(a\right)+T\left(b\right)$.

3. $T\left(ca\right)=cT\left(a\right)$Where c is any scalar and $a\in {R}^{m}$

## Step 3: Checking for linear transformation

Identity of R3 is (0,0,0).

Now we will find the transformation of identity element.

$T\left(0,0,0\right)=\left(2\left(0\right),0+2,2\left(0\right)\right)=\left(0,2,0\right)\ne \left(0,0,0\right)$

First condition of linear transformation does not hold.

Thus, it is not a linear transformation.