How to do the following change of Basis

Question

$V=\mathbb{Q}^3$ The two following Bases of V are given

$S=\{(1,0,0) , (0,1,0) , (0,0,1)\}, T=\{(-4,2,1) , (-1,0,1), (1,-1,1)\}$;

Let $w ∈ V$ with $γ_T$$(w)$ =$ $$\begin{pmatrix}1\\0\\1\end{pmatrix}$. Determine $γ_S(w)$

Let $v ∈ V$ with $γ_S$$(v)$ =$ $$\begin{pmatrix}3\\-2\\1\end{pmatrix}$. Determine $γ_T(v)$

"How would I begin to solve this problem/what steps should be taken to solve it."

Answer 1

Hint: You need to find the change of basis matrix. For the $T \rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.

Answer 2

$\gamma_T \rightarrow \gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $\gamma_S(w)$ you simply add the components of these vectors.

$\gamma_T \rightarrow \gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $\gamma_S$ components to $\gamma_T$ components.