Trying to review for a quiz and I'm woefully underprepared.
Question is:
For each of the following linear transformations $L$ mapping $\mathbb{R}^3$ into $\mathbb{R}^2$, find a matrix $A$ such that $L(x) = Ax$ for every $x$ in $\mathbb{R}^3$.
a) $L((x_1, x_2, x_3)^T) = (x_1 + x_2, 0)^T$
I don't quite know how to approach this problem. It's probably very simple, but the change from $\mathbb{R}^3 \to \mathbb{R}^2$ or from $\mathbb{R}^2 \to \mathbb{R}^3$ confuses me, I'm not sure how to treat it.
I'm looking for a slow demonstration of steps rather than an actual answer. Calculations are nice, but I'd rather learn the steps and reasoning behind them so I can complete these types of questions on my own later.
Best Answer
Since the function takes in a triple and outputs a double, you are looking for a $2$x$3$ matrix such that $$\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{bmatrix}\cdot \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}=\begin{bmatrix} x_1+x_2 \\ 0 \end{bmatrix}$$
So we have $$a_1\cdot x_1 + b_1 \cdot x_2 + c_1 \cdot x_3 = x_1+x_2$$ $$a_2\cdot x_1 + b_2 \cdot x_2 + c_2 \cdot x_3 =0$$
Therefore, $a_1=1, b_1=1$ and the rest equal $0$.