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Q34E

Expert-verifiedFound in: Page 54

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Consider the transformation T from ${{\mathit{R}}}^{{\mathbf{2}}}$**

Matrix for T will be $\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$

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

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

Given T is linear transformation from${R}^{2}\to {R}^{2}$.

Let be any point of${R}^{2}$.

Then we can write the transformation.

$\begin{array}{rcl}T(r\mathrm{cos}\alpha ,r\mathrm{sin}\alpha )& =& \left(r\mathrm{cos}\right(\alpha +\theta ),r\mathrm{sin}(\alpha +\theta \left)\right)\\ & =& (r\mathrm{cos}\alpha \mathrm{cos}\theta -r\mathrm{sin}\alpha \mathrm{sin}\theta ,r\mathrm{sin}\theta \mathrm{cos}\alpha +r\mathrm{cos}\theta \mathrm{sin}\alpha )\\ & =& \left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}r\mathrm{cos}\alpha \\ r\mathrm{sin}\alpha \end{array}\right]\\ {\left[T\right]}_{\beta}& =& \left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\end{array}$

Hence, the matrix for T will be $\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$.

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