# Find the inverse of a transformation

linear algebralinear programmingmatrices

I am studying pure maths as a hobby. I am trying to understand inverse transformations. The text book I am using gives an example of how to do it:

Find the inverse transformation $$T^{-1}$$ of $$T:\begin{pmatrix}x\\y\\z\end{pmatrix}\mapsto \begin{pmatrix}x\\x+y\\z\end{pmatrix}$$

Let $$\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\end{pmatrix}=\begin{pmatrix}x\\x+y\\z\end{pmatrix}$$

Then: $$x=p_{1}, z=p_{3}, y=p_{2}-x=p_{2}-p_{1}$$

So $$T^{-1}: \begin{pmatrix}p_{1}\\p_{2}\\p_{3}\end{pmatrix}\mapsto \begin{pmatrix}p_{1}\\ p_{2}-p_{1}\\ p_{3}\end{pmatrix}$$

or $$T^{-1}: \begin{pmatrix}x\\y\\z\end{pmatrix}\mapsto \begin{pmatrix}x\\ y-x\\ z\end{pmatrix}$$

An end of section question asks: Find $$T^{-1}$$ in the case where $$T: \begin{pmatrix}x\\y\\z\end{pmatrix}\mapsto \begin{pmatrix}x+y\\ x+y+z\\ 2y-z\end{pmatrix}$$

But I cannot successfully apply this technique. I have said:

$$\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\end{pmatrix}=\begin{pmatrix}x+y\\x+y+z\\2y-z\end{pmatrix}$$

$$x+y=p_{1}, x+y+z=p_{2}, 2y-z=p_{3}$$

Presumably I am meant to find a RHS consisting entirely of $$p_{1},p_{2}$$ and $$p_{3}$$ but I cannot find a solution to this.

You have the system$$\left\{\begin{array}{l}x+y=p_1\\x+y+z=p_2\\2y-z=p_3.\end{array}\right.$$If you replace the second equation by the second equation minus the first one, it becomes$$\left\{\begin{array}{l}x+y=p_1\\z=p_2-p_1\\2y-z=p_3.\end{array}\right.$$So, now you know that $$z=p_2-p_1$$. It follows now from the third equation that $$y=\frac12(-p_1+p_2+p_3)$$. And now you can use the first equation to get $$x$$.