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 teh 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 &amp; 1\\<br /> -1 &amp; 0<br /> \end{array}<br /> \right]