MATH 251 Lecture 7

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Transformations

Translation

Add a vector $\overrightarrow{b}$, and the graph is shifted in the direction of $\overrightarrow{b}$

Polar Coordinates

Whoa! What just happened?

It turns a line into a circle!

Linear Transformation

Point $\overrightarrow{X}=\left(\begin{array}{c}x\\ y\end{array}\right)$ maps to ($\mapsto$) $A\overrightarrow{X}$:

$\begin{array}{rl}A& =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\\ A\overrightarrow{X}& =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}ax+by\\ cx+dy\end{array}\right)& =x\left(\begin{array}{c}a\\ c\end{array}\right)+y\left(\begin{array}{c}b\\ d\end{array}\right)\end{array}$

The resulting vectors $\overrightarrow{v}=⟨a,c⟩$ and $\overrightarrow{w}=⟨b,d⟩$ create a "transformed lattice" as compared to the original lattice made by $\hat{ı}$ and $\hat{ȷ}$

According to the formula,

$\left(\begin{array}{c}x\\ y\end{array}\right)=x\hat{ı}+y\hat{ȷ}\mapsto x\overrightarrow{v}+y\overrightarrow{w}$

Area of Linear Transformation

Calculate the area of a shape in $\hat{ı}$ and $\hat{ȷ}$ lattice, then multiply by the absolute value of the determinant of the transform matrix:

$\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}=\left|\mathrm{d}\mathrm{e}\mathrm{t}\left(A\right)\right|=|ad-bc|$

Rotations

Just rotate the lattice vectors by an angle $\theta$:

$\begin{array}{rl}\hat{ı}& \mapsto \left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)\\ \hat{ȷ}& \mapsto \left(\begin{array}{c}-\sin \theta \\ cos\theta \end{array}\right)\end{array}$

Therefore the translation matrix is:

$A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$