MATH 251 Lecture 7

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Transformations

Translation

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

Polar Coordinates

Whoa! What just happened?

It turns a line into a circle!

Linear Transformation

Point ${\vec {X}}={\begin{pmatrix}x\\y\end{pmatrix}}$ maps to ($\mapsto$) $A{\vec {X}}$:

${\begin{aligned}A&={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\\A{\vec {X}}&={\begin{pmatrix}a&b\\c&d\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}}&=x{\begin{pmatrix}a\\c\end{pmatrix}}+y{\begin{pmatrix}b\\d\end{pmatrix}}\end{aligned}}$

The resulting vectors ${\vec {v}}=\left\langle a,c\right\rangle$ and ${\vec {w}}=\left\langle b,d\right\rangle$ create a "transformed lattice" as compared to the original lattice made by ${\hat {\imath }}$ and ${\hat {\jmath }}$

According to the formula,

${\begin{pmatrix}x\\y\end{pmatrix}}=x{\hat {\imath }}+y{\hat {\jmath }}\mapsto x{\vec {v}}+y{\vec {w}}$

Area of Linear Transformation

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

$\mathrm {Area} =\left|\mathrm {det} \left(A\right)\right|=\left|ad-bc\right|$

Rotations

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

${\begin{aligned}{\hat {\imath }}&\mapsto {\begin{pmatrix}\cos {\theta }\\\sin {\theta }\end{pmatrix}}\\{\hat {\jmath }}&\mapsto {\begin{pmatrix}-\sin {\theta }\\cos{\theta }\end{pmatrix}}\end{aligned}}$

Therefore the translation matrix is:

$A={\begin{pmatrix}\cos {\theta }&-\sin {\theta }\\\sin {\theta }&\cos {\theta }\end{pmatrix}}$