Let $\beta =\{b_1,b_2,b_3\}$ be a basis for the vector space V and let $T:V\to R^2$ be a transformation with the property that $$T(x_1b_1 + x_2b_x + x_3b_3) = \begin{bmatrix} 2x_1-3x_2+x_3\\ -2x_1+5x_3\end{bmatrix}$$ Find the matrix for T relative to $\beta$ and the standard matrix for T

I found $[T]_\beta$ $$[T]_\beta=\begin{bmatrix} 2 & -3 & 1\\ -2& 0 & 5\\ \end{bmatrix}$$ The next thing I need to do to find the matrix T relative to the standard matrix. To find $[T]_\beta$ I analised $T(x_1b_1 + x_2b_x + x_3b_3)$ into $$\begin{bmatrix} T(b_1) & T(b_2) & T(b_3)\end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3\\ \end{bmatrix}$$ where $$\begin{bmatrix} T(b_1) & T(b_2) & T(b_3)\end{bmatrix} = [T]_\beta$$

Now, what I need to do to find the T relative to the standard basis for $R^2$. Please guide me.


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