If $L(x_1)$ = $L(x_2)$, does $x_1$ = $x_2$? Reviewing for an exam, and the book says False. But I can't seem think of a way to prove this wrong...


  If $L(x_1)$ = $L(x_2)$, does $x_1$ = $x_2$?


where $L$ is a linear transformation from $\Bbb R^n$ to $\Bbb R^n$, and $x_1$ and $x_2$ are vectors.
A trivial example would be the zero transformation
$$L: \mathbb{R}^n \to \mathbb{R}^n, L(x) = 0 \; \forall x \in \mathbb{R}^n.$$
Then $e_1 \neq e_2$ but $L(e_1) = 0 = L(e_2)$.

In general, not only in the context of vector spaces, $L(x_1) = L(x_2) \implies x_1 = x_2$ if and only if $L$ is injective. If, in particular, $L$ is a linear operator on vector spaces, then $L(x_1) = L(x_2) \implies x_1 = x_2$ if and only if $\text{ker}(L) = \{0\}$.
Consider x1 = -x2, and and L transforms the elements in a vector into absolute value. Then L(x1) = L(x2) even though x1 is not equal to x2.