Linear Algebra Proof of Mean Value Theorem

It's not valid that you can do this; when you rotate the graph of a function, it may no longer be a function. I think it's a good exercise to come up with an example where this happens.

However, you can prove the mean value theorem from Rolle's theorem: define a new function to be equal to $f$ minus the line through $(a,f(a))$ and $(b,f(b))$ and then apply Rolle's Theorem.

If the function is concave downward everywhere and has a vertical tangent at both endpoints, then this method will not work. Draw a picture and you'll see why. In fact, if the tangent is steep enough but not vertical, this won't work because the resulting graph will fail the vertical line test.

The linear transformation whose matrix is $$ \begin{bmatrix} 1 & 0 \\ -m & 1 \end{bmatrix}, $$ where $m$ is the slope of the secant line, will work. And that is just what is usually done, even if it's not phrased in the language of linear algebra and matrices.