how to begin to find a matrix that is 2x2 for this question.

uplakanimkk
2022-07-14
Answered

Suppose $T$ is a transformation from ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$. Find the matrix $A$ that induces $T$ if $T$ is the (counter-clockwise) rotation by $\frac{3}{4}}\pi $.

how to begin to find a matrix that is 2x2 for this question.

how to begin to find a matrix that is 2x2 for this question.

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Dayana Zuniga

Answered 2022-07-15
Author has **16** answers

The rotation matrix is $\left[\begin{array}{cc}\mathrm{cos}(\theta )& -\mathrm{sin}(\theta )\\ \mathrm{sin}(\theta )& \mathrm{cos}(\theta )\end{array}\right]$But blindly applying this formula doesn't teach you the most important part of linear transformations. (ANY linear transformation matrix is defined by where it takes the unit vectors.)

This linear transformation T that is a counterclockwise rotation takes $\left[\begin{array}{c}1\\ 0\end{array}\right]$ to $\left[\begin{array}{c}\mathrm{cos}(\theta )\\ \mathrm{sin}(\theta )\end{array}\right]$ and $\left[\begin{array}{c}0\\ 1\end{array}\right]$ to $\left[\begin{array}{c}-\mathrm{sin}(\theta )\\ \mathrm{cos}(\theta )\end{array}\right]$. You can confirm this on your own using geometry.

And this works for ANY linear transformation.

This linear transformation T that is a counterclockwise rotation takes $\left[\begin{array}{c}1\\ 0\end{array}\right]$ to $\left[\begin{array}{c}\mathrm{cos}(\theta )\\ \mathrm{sin}(\theta )\end{array}\right]$ and $\left[\begin{array}{c}0\\ 1\end{array}\right]$ to $\left[\begin{array}{c}-\mathrm{sin}(\theta )\\ \mathrm{cos}(\theta )\end{array}\right]$. You can confirm this on your own using geometry.

And this works for ANY linear transformation.

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Rotate a triangle with vertices (2, 1). (4. 1). (3, 3) by 90 degrees (clock wise) and then Translate it by Tx 4 and Ty 3 and finally rotate it by 30 degrees (anti-clock wise). (cos90=0, sin90 1. cos 30 = 0.866, sin30 =0.5)

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What is correct order of transformation matrix to multiply the metrix?

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What is correct order of transformation matrix to multiply the metrix?

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Should i follow the order given by the question or reverse order?

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