Condition for Mapping from Quotient Vector Space to be Well-Defined
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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 $L \circ Q_M = T$
- $M \subseteq \ker T$
- $\begin {xy} \xymatrix@L + 2mu@ + 1em {V \ar[r]^*{T} \ar[d]_*{Q_M} & W \\ V / M \ar@{-->}[ur]_*{L} } \end {xy}$
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 $v + M = v' + M$ 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 $M \subseteq \ker T$.
$\blacksquare$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra: Proposition $1.3$