Suppose $v_1,\dots,v_m$ is a list of vectors in $V$. Define $T \in \mathcal L(\mathbf F^n, V)$ by

1. What property of $T$ corresponds to $v_1,\dots,v_m$ spanning $V$?
2. What property of $T$ corresponds to $v_1,\dots,v_m$ being linearly independent?

Note $\text{range }T$ denotes the span of $v_1,\dots,v_m$ and $\text{null }T$ denotes all linear combinations giving zero

1. $T$ being surjective, or equivalently $\text{range }T = V$
2. $T$ being injective, or corresponds $\text{null }T = \{0\}$ (no combinations give zero)