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 $L \circ Q_M = T$

$M \subseteq \ker T$

$\begin {xy} \xymatrix@L + 2mu@ + 1em {V \ar[r]^*{T} \ar[d]_*{Q_M} & W \\ V / M \ar@{-->}[ur]_*{L} } \end {xy}$

$L: V / M \to W$ exists

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

Now $v + M = v' + M$ if and only if $v - v' \in M$.

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 $M \subseteq \ker T$.

$\blacksquare$

1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra: Proposition $1.3$