Describing $T(\vec v)$ for all $\vec v$ in arbitrary finite-dimensional vector space $V$

Question

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?

Answer 1

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 $T \left( x_1, x_2, x_3, x_4 \right) = \left( x_1 + x_3, x_2 + x_4, x_1 + x_3 \right)$.

Answer 2

It is basically the same answer that Aniruddha already gave us, I just want to present it slightly differently.

Let $\vec{v}$ be an arbitrary vector in $\mathbb{R}^4$. Then there exist unique real coefficients $c_1, c_2, c_3, c_4$ such that $$\vec{v} = c_1\vec{v}_1 + c_2\vec{v}_2 + c_3\vec{v}_3 + c_4 \vec{v}_4.$$

Your representation matrix $\operatorname{Mat}_{\mathscr{C,B}}(T) $ tells you that $T(\vec{v}_i)$ is the $i$-th column of $\operatorname{Mat}_{\mathscr{C,B}}(T)$ (where $i=1,2,3,4$).

Therefore, we have $$\begin{align} T(\vec{v}) &= T(c_1\vec{v}_1 + c_2\vec{v}_2 + c_3\vec{v}_3 + c_4 \vec{v}_4) \\[0.2cm] &= c_1 T(\vec{v}_1) + c_2T(\vec{v}_2) + c_3T(\vec{v}_3) + c_4 T(\vec{v}_4) \\[0.2cm] &= c_1 \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + c_2 \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + c_3 \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + c_4 \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \\[0.2cm] &= \begin{pmatrix} c_1 + c_3 \\ c_2 + c_4 \\ c_1 + c_3 \end{pmatrix} \end{align} $$ which is probably the required description in your exercise.