Quotient spaces and composition of linear maps

Question

Let $V,W$ be vector spaces over a field $\mathbb{F}$, $f:V\to W$ a linear map and $V'\subset V$, $W'\subset W$. Show that $f(V')\subset W'$ iff there exists a linear map $F:V/V'\to W/W'$, such that $F\circ \pi_V=\pi_w\circ f$ where $\pi_V,\pi_W$ denote the canonical linear maps to the respective quotient spaces.


I've been stuck with this for a while and would greatly appreciate any help. Thank you very much in advance.

Answer 1

Suppose $F$ exists. If $v\in V’$, then $\pi_V(v)=\mathbf{0}$, so the left hand side of $F\circ \pi_V$ evaluates to $\mathbf{0}$. Hence, so does the right hand side, so $\pi_W(f(\mathbf{v}))=\mathbf{0}$. Thus, $f(\mathbf{v})\in\mathrm{ker}(\pi_W)$.

Conversely, if $f(V’)\subseteq W’$, define $\mathfrak{F}\colon V\to W/W’$ by $\pi_W\circ f$. Show that $V’\subseteq\mathfrak{F}$, so this induces the map $F$.