# How to formally and explicitly verify the last statement of the following?

I am trying to verify the final statement in the following picture. The bases in example 1 are $$\beta = \{(1,1), (1, -1) \}$$ and $$\beta' = \{(2,4), (3, 1) \}$$

What exactly is $$[T]_{\beta}$$? From my understanding it's the identity matrix of space generated by T with respect to the basis $$\beta$$.

Taken from the 4th edition Linear Algebra by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence. Page 128

$$$$\begin{split} [T]_\beta &= \beta^{-1} T \beta \\ &=\begin{bmatrix} 1 && 1 \\ 1 && -1 \end{bmatrix}^{-1} \begin{bmatrix} 3 && -1 \\ 1 && 3 \end{bmatrix} \begin{bmatrix} 1 && 1 \\ 1 && - 1\end{bmatrix} \\ &= -\frac{1}{2} \begin{bmatrix} -1 && -1 \\ -1 && 1 \end{bmatrix} \begin{bmatrix} 2 && 4 \\ 4 && -2 \end{bmatrix} \\ &= \begin{bmatrix} 3 && 1 \\ -1 && 3 \end{bmatrix} \end{split}$$$$
The $$[T]_{\beta}$$ denotes the matrix associated with the linear transformation $$T$$ with respect to the ordered basis $$\beta$$, and satisfies $$[Tv]_{\beta}=[T]_{\beta}[v]_{\beta}$$ for all vectors $$v \in \mathbb{R^2}$$. If $$\beta_1=(1,1)$$ and $$\beta_2=(1,-1)$$, then $$[T]_{\beta}=[A_1,A_2]$$ where $$A_i=[T\beta_i]_{\beta}, i=1,2.$$ You can find the column vectors $$A_i$$ of the matrix $$[T]_{\beta}$$ by evaluating the basis vectors $$\beta_i$$ with respect to $$T$$. The result is $$T\beta_1=3\beta_1-\beta_2$$ and $$T\beta_2=\beta_1+3\beta_2$$.