Basis maps to Linearly Independent Set How do I prove that a linear map f will map a basis to a linearly independent set?
Supposedly this is examinable but damned if it's not on either set of notes.
Thanks!

Suppose that $T:V\rightarrow W$ is a injective linear map. Let $\left\{v_1,\dots ,v_n\right\}$ be a basis of $V$ (You can do a similar argument if $V$ is infinite-dimensional). Suppose that $\sum_{i=1}^n\lambda_iT(v_i)=0$, then $T(\sum_{i=1}^n\lambda_iv_i)=0$. Hence $\sum_{i=1}^n\lambda_iv_i\in \text{Ker}(T)$, but since $T$ is injective, $\text{Ker}(T)=\left\{0\right\}$. Hence $\sum_{i=1}^n\lambda_iv_i=0$. Since $\left\{v_1,\dots , v_n\right\}$ is a basis, we need that $\lambda_i=0$ for all $i$. Hence the $T(v_i)$'s are linearly independent.
