Linear transformation from $\mathbb R^2$ to $\mathbb M_2$

Question

This is my first question here. I'm studying linear algebra and I don't know how to do this exercise:

Determine T: $\mathbb{R^{2}} \to M_2 (\mathbb{R})$ knowing that T(1,0) = $$\begin{bmatrix} 1 & -1 \\ 2 & 0 \end{bmatrix}$$ and T(0,1) = $$\begin{bmatrix} -2 & 0 \\ 1 & 1 \end{bmatrix}$$

Can you help me doing this?

Answer 1

If $(x,y)\in\mathbb R^2$, then\begin{align}T(x,y)&=T\bigl(x(1,0)+y(0,1)\bigr)\\&=xT(1,0)+yT(0,1)\\&=\begin{bmatrix}x-2y&-x\\2x+y&y\end{bmatrix}.\end{align}

Answer 2

Use linearity of $T$ and the fact that $(a,b)=a(1,0)+b(0,1)$

To get

$$T(a,b)=aT(1,0)+bT(0,1)$$