It seems I have fallen behind in my Methods of Theoretical Physics class, and am asking here as I need explanations and resources to learn. Let's say we are given the following:
"Find the transformation matrix, $N$, which stretches vectors by a factor of 4 along the line $y = 2x$."
How would you go about this? If this just means "the matrix that stretches the y-axis by a factor of 4 and rotates such that $y' = 2x$", then I think I can do that, but would rather the help to be sure.
Best Answer
Choose a basis for the coorinate system.
$e_1 = \begin {bmatrix} 1\\2\end{bmatrix}, e_2=\begin{bmatrix}-2\\1 \end{bmatrix}$
That is one vector in the direction of $y=2x$ and one orthogonal.
$B = \begin{bmatrix} 1&-2\\2&1\end{bmatrix}$ will transform a vector in our basis to the standard basis.
$B^{-1}$ will then transform a vector in the standard basis to our basis.
$\begin{bmatrix} 4\\&1\end{bmatrix}$ will transform a vector in our basis as desired.
$B\begin{bmatrix} 4\\&1\end{bmatrix}B^{-1}$ will take a vector in the standard basis, transform it to our adjusted frame of reference, dilate in one direction, leaving the other unchanged, and transform back to the standard basis.
$\begin{bmatrix} 1&-2\\2&1\end{bmatrix}\begin{bmatrix} 4\\&1\end{bmatrix}\begin{bmatrix} \frac15&\frac 25\\-\frac25&\frac 15\end{bmatrix}= \begin{bmatrix}\frac 85&\frac 65\\\frac 65&\frac {17}5\end{bmatrix}$