My answer to the following exercise seems dangerously simplistic.
Since I must describe how vectors in $\mathbb R^4$ are mapped to $\mathbb R^3$ through this linear transformation, I merely looked at the matrix of $T$ given to me and stated:
$T\begin{pmatrix}
x \\
y \\
z \\
t \\
\end{pmatrix} = T
\begin{pmatrix}
x+y \\
y+t \\
x+z \\
\end{pmatrix}$
However, I feel like this was too simple. An exercise wouldn't have me just read the coefficents off of a matrix. Is this truly all I had to do, and if not, what instead?
Best Answer
Considering that there was a typing mistake in your question and you wanted to have the linear transformation $T: \mathbb{R}^4 \rightarrow \mathbb{R}^3$, we just need to check what happens when $T$ operates on any abitrary vector $\left( x_1, x_2, x_3, x_4 \right) \in \mathbb{R}^4$. It is same as saying that what happens when the matrix $\text{Mat}_{\mathscr{C}, \mathscr{B}} \left( T \right)$ operates on the column vector $\left[ \begin{matrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{matrix} \right]$. So, we have
$$\left[ \begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{matrix} \right] \left[ \begin{matrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{matrix} \right] = \left[ \begin{matrix} x_1 + x_3 \\ x_2 + x_4 \\ x_1 + x_3 \end{matrix} \right]$$
Thus, the action of $T$ on any arbitrary vector $v \in \mathbb{R}^4$ is given by $T \left( x_1, x_2, x_3, x_4 \right) = \left( x_1 + x_3, x_2 + x_4, x_1 + x_3 \right)$.