Given that $T: V \to W$ is a linear transformation from $V$ to $W$. Show that if $T$ is surjective and $S\subset V$ spans $V$, then $T(S)$ spans $W$.
I think the main thing stumping me right now is how to use spanning to show anything or what I would need to prove in order to show that $W$ has been successfully spanned. I understand how to use the definition of linear transformation and surjective, but I can't even get started (OR CAN I???) without a clear understanding of how spanning works.
I understand span in concrete numbers, so I need help getting a more abstract concept of $\operatorname{span} (S)$, if possible. And I tried YouTube and reading definitions already, I am truly stuck.
Best Answer
$S$ spans $V$ simply means that every element $v \in V$ can be written (not necessarily uniquely) as a finite linear combination of elements of $S$.
so it is required to prove that every element $w \in W$ can be written as a finite linear combination of elements of $T(S)$
now, the fact that $T$ is surjective means that any $w \in W$ is the image (under the mapping $T$) of some (not necessarily unique) element of $V$. i.e. we can find $u \in V$ such that:
$$ w = T(u) $$
write $u$ as a finite linear combination of elements of $S$:
$$ u = \sum_k \lambda_k s_k $$ where the $\lambda_k$ are scalars and each $s_k \in S$
since the mapping $T$ is linear we have:
$$ w = T(u) = T\bigg(\sum_k \lambda_k s_k \bigg) = \sum_k \lambda_k T(s_k) $$ showing that $w$ is a linear combination of elements of $T(S)$ as required.