MATH 240

Mon. September 16th, 2019

Rotation Matrix

(cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}
Rotates a vector by θ\theta degrees counter-clockwise.

Projection Matrices

(100000001)\begin{pmatrix} 1&0&0\\0&0&0\\0&0&1 \end{pmatrix}
Projects a vector onto the xx-zz plane (the yy component becomes 0).

Linear Transformations

The point of showing these matrices is that there are ways to think about multiplying vectors by matrices outside the context of linear systems of equations. (These matrices aren’t always in row-reduced form.)

These can be thought of as functions (for every input there is one output) with special properties.

T: Rn→RmT(x+y)=T(x)+T(y)T(cx)=cT(x)T(0)=0\begin{aligned} T: \ &\R^n \rightarrow \R^m\\ T(x+y)&=T(x)+T(y)\\ T(cx)&=cT(x)\\ T(0)&=0 \end{aligned}
(This third property shows that the function has a fixed origin, and is a result of the first property.)

We call any function with these properties a linear transformation, even if it’s not a matrix.

For all rr in R\R, Tr(x)=rxT_r(x) = rx is a linear transformation.
If 0<r<10<r<1, then TrT_r is called a contraction.
If r>1r>1, then TrT_r is called a dialation.

If AA is a m×nm\times n matrix, then T(x)=AxT(x)=Ax is a linear transformation from Rn\R^n to Rm\R^m.