Find the 2 x 2 rotation dilation matrix which rotates by 45◦ counter clockwise and scales by a factor $\sqrt{8}$.
I have no idea how to do this. Our teacher literally did not explain this at all. I have done so much research and have NOT found a single thing that can guide me through this. Can someone please help me out? PLEASE?
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A very important fact to remember for solving problems like this one is that the columns of a transformation matrix are the images of the basis vectors. If you can’t think of any other way to construct a particular matrix, you can always fall back on this principle. If the matrix is supposed to be given relative to the standard basis, then work out the results of applying the transformation to the standard basis and you’ve got your matrix.
The matrix which rotates a 2-dimensional vector through some angle $\theta$ is $$\begin{bmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{bmatrix},$$ and the matrix that scales an $n$-dimensional vector by a factor of $\lambda$ is given by $\lambda I_n$, where $I_n$ is the $n\times n$ identity matrix.
To produce one matrix with multiple translational effects, you simply multiply the matrices corresponding to the individual effects. Can you take it from here?