For a linear map $A\in Hom(U,V)$ and a linearly independent subset $L\subseteq U$ where U and V are vector spaces, what does the following statement mean:
$A$ is injective $\left.\Rightarrow A\right|_{\text {Span } L}$ is injective $\left.\Leftrightarrow A\right|_{L}$ is injective and $A(L)$ is linearly independent.
I do not know what the notation $\left.A\right|_{\text {Span } L}$ means and therefore don't understand the statement.
Since $L \subset U$, one can talk of the linear span of $L$. The span of a subset is the smallest subspace that contains the given set. The span of a subset always exists(it is the intersection of all the subspaces that contain the given set). The span of a subset $S$ is usually denoted $span(S)$. Since $A$ is a linear transformation on $A$, you can restrict it to the subspace $span(L)$ and this new linear transformation is denoted by $A|_{span(L)}$.