I'm working through Hoffman's linear algebra book right now for a proof-based class on Linear Algebra and I'm a little confused about some of the calculations regarding linear transformations.
For this question in particular, I am not sure if I have done it correctly or how to do part (c),
Let $T$ be the linear operation on $\mathbb{R}^3$ defined by, $T(x_{1},x_{2},x_{3})=(3x_{1} + x_{3},-2x_{1} + x_{2}, -x_{1} + 2x_{2} + 4x_{3})$ $(a)$ What is the matrix of $T$ in the standard ordered basis for $\mathbb{R}^3$?
$(b)$ What is the matrix of $T$ in the ordered basis $\{\alpha_{1},\alpha_{2},\alpha_{3}\}$, where $\alpha_{1}=(1,0,1), \alpha_{2}=(-1,2,1),$ and $\alpha_{3}=(2,1,1)$?
$(c)$ Prove that $T$ is invertible and give a rule for $T^{-1}$ like the one which defines $T$.
For $(a)$, I think you simply compute $T(1,0,0), T(0,1,0)$, and $T(0,0,1)$ and then whatever you get will be the entries in $[T]_{\mathbb{B}}$, where $\mathbb{B}$ denotes the standard basis for $\mathbb{R}^3$. That is,
$[T]_{\mathbb{B}} =\left[ \begin{array}{ccc} 3 & -2 & -1 \\ 0 & 1 & 2 \\ 1 & 0 & 4 \\ \end{array} \right]$
Now, for $(b)$ I am pretty sure that you just compute $T(1,0,1), T(-1,2,1),$ and $T(2,1,1)$ to find the matrix of $T$ in the ordered basis $\{\alpha_{1},\alpha_{2},\alpha_{3}\}$. Then, we get for $1 \leq i \leq 3$,
$[T]_{\mathbb{\alpha_{i}}} =\left[ \begin{array}{ccc} 4 & -2 & 3 \\ -2 & 4 & 9 \\ 7 & -3 & 4 \\ \end{array} \right]$
Let me know if I have gone wrong so far. My idea for $(c)$ was to invert this matrix and then somehow deduce a rule from there, but I am not sure.
My question is: How can I answer (c)? (and also, if I have done anything wrong thus far let me know).
This is not homework.