Condition for Mapping from Quotient Vector Space to be Well-Defined

Theorem
Let $V, W$ be vector spaces.

Let $T: V \to W$ be a linear transformation.

Let $M$ be a subspace of $V$.

Let $V / M$ be the quotient vector space of $V$ by $M$.

Let $Q_M: V \to V / M$ be the associated quotient mapping.

Then:
 * there exists a linear transformation $L: V / M \to W$ such that $L \circ Q_M = T$


 * $M \subseteq \ker T$
 * $M \subseteq \ker T$


 * $\begin {xy} \xymatrix@L + 2mu@ + 1em {

V \ar[r]^*{T} \ar[d]_*{Q_M} & W \\ V / M \ar@{-->}[ur]_*{L} } \end {xy}$

Proof
By Condition for Mapping from Quotient Set to be Well-Defined, it follows that:


 * $L: V / M \to W$ exists


 * $\forall v, v' \in V: v + M = v' + M \implies T v = T v'$

Now $v + M = v' + M$ $v - v' \in M$.

Since $T$ is linear:


 * $T v = T v' \iff T \paren{ v - v' } = 0$

In particular then, with $v' = 0$ it follows that:


 * $L: V / M \to W$ exists


 * $\forall v \in V: v \in M \implies T v = 0$

and the latter states that $M \subseteq \ker T$.