Check work on linear transformation problem

[tandoorichicken]

check work please on linear transformation problem

The problem is to find a standard matrix of T.
$$T:\mathbb{R}^3\rightarrow\mathbb{R}^2, T(\vec{e}_1) = (1,3), T(\vec{e}_2) = (4,-7), T(\vec{e}_3) = (-5,4)$$
where e_1, e_2, and e_3 are the columns of the 3x3 identity matrix.
So here's what I did:
Find A for $T(\vec{x})=A\vec{x}$

$$\vec{x}=I_3\vec{x}=\left[ \vec{e}_1 \; \vec{e}_2 \; \vec{e}_3 \right] \vec{x} = x_1\vec{e}_1 + x_2\vec{e}_2 + x_3\vec{e}_3$$

$$T(\vec{x}) = T(x_1\vec{e}_1 + x_2\vec{e}_2 + x_3\vec{e}_3) = x_1 T(\vec{e}_1) + x_2 T(\vec{e}_2) + x_3 T(\vec{e}_3)$$

$$\left[ T(\vec{e}_1) \; T(\vec{e}_2) \; T(\vec{e}_3)\right]\vec{x} = A\vec{x}$$

$$A = \left[ T(\vec{e}_1) \; T(\vec{e}_2) \; T(\vec{e}_3)\right] =$$
(well basically i screwed up the tex for this but its a 2x3 matrix with top row 1,4,-5 and bottom row 3,-7,4)
Note: a lot of this might be unnecessary, but my main goal is that I want to be sure that I am understanding this correctly.

 

[tandoorichicken]

I think I finally understand linear transformation. Can someone check this one also?

$T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ first reflects points through the horizontal $x_1$-axis and then reflects points over the line $x_2=x_1$

what I did:

Find A for $T(\vec{x})=A\vec{x}$

Let $T_1$ be the first reflection.
$$\vec{e}_1 = (1,0), \vec{e}_2 = (0,1)$$
$$T_1(\vec{e}_1) = (1,0), T_1(\vec{e}_2) = (0,-1)$$

Let $T_2$ be the second reflection.
$$T(\vec{e}_1) = T_2(T_1(\vec{e}_1)) = (0,1), T(\vec{e}_2) = T_2(T_1(\vec{e}_2)) = (-1,0)$$
$$A = [T(\vec{e}_1) \; T(\vec{e}_2)] = \left[<br /> \begin{array}{cc}<br /> 0 & 1\\<br /> -1 & 0<br /> \end{array}<br /> \right]$$