I've been asked to find a matrix representing projection of a point in the plane onto the line $ax + by = 0.$
I know the matrix for $ax+by$ is $$\left[\begin{array}{cc}a &0\\ b&0\end{array}\right]$$ I also know projection of $u$ onto $v$ is $((u\cdot v)/\|v\|^2)v$. What would I be projecting onto?
A second question is finding a matrix for reflection about the line $4x + 5y = 0$ followed by rotation by $\pi/2$ radians clockwise about the origin. Matrix: $$\left[\begin{array}{cc}4& 0\\ 5& 0\end{array}\right].$$ Where should I go from here?
For writing the matrix of any transformation, the columns $[a_1 a_2 ... a_n]$ are where the unit vectors $e_1,e_2,...e_n$ are mapped to by the transformation. For your first question, we are projecting onto the line $ax+by=0$,which is also the span of the vector $$\begin{pmatrix}-\frac{1}{a}\\\frac{1}{b}\end{pmatrix}$$ So, to find the transformation we calculate the image of $e_1$ and $e_2$ when they are projected onto this span, and take those new coordinates to be the columns of the matrix.
For the second one, it is slightly more complicated. Note that the projection of a vector onto a span is halfway between the original vector and the reflected vector. Letting $\vec P$, $\vec R$, $\vec x$ be the projected vector, reflected vector and original vector respectively, we have $ \vec P =\frac{1}{2}(\vec R +\vec x)$, or $\vec R=2\vec P-\vec X.$ This can be translated into matrices by taking the matrix of the projection for $\vec P$, the matrix of the reflection for $\vec R$, and the identity matrix for $\vec X$.This gives us $[R]=2[P]-[I]$.
Applying this to $e_1$ and $e_2$ will give the columns for our reflection matrix, and now we apply the rotation matrix, $$\begin{pmatrix}\cos(\theta) &-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{pmatrix}$$ where $\theta$ is the angle of rotation. You will plug in not $e_1$ and $e_2$, but their images under the initial transformation, and their new images under this transformation will be the columns of your new matrix.
Note: For the second part, an alternate method would be to find the matrix of the reflection and rotation individually, and then use matrix multiplication as it corresponds to the composition of functions.