TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Universal property of the quotient vector space

Statement

Let $f : V \to U$ be linear and let $W \leq V$ satisfy $W \subseteq \ker f$ . There is a unique linear map $\overset{ˉ}{f} : V / W \to U$ such that $\overset{ˉ}{f} \circ π = f .$ It is given by $\overset{ˉ}{f} (v + W) = f (v)$ .

Facts & Assumptions

Given: A linear map $f : V \to U$ and a subspace $W \subseteq \ker f$ .

[L1]

$v + W = v^{'} + W$ exactly when $v - v^{'} \in W$ , the operations on $V / W$ make it a vector space over $F$ , and the canonical projection $π : V \to V / W$ is a surjective linear map with $\ker π = W$ (Coset equality, well-defined quotient operations, and the canonical projection with kernel $W$ ).

[L2]

For a linear map $f$ , its kernel is $\ker f = {v : f (v) = 0}$ (Kernel and image of a linear map).

[L3]

The quotient operations are $(v + W) + (u + W) : = (v + u) + W$ and $a (v + W) : = (a v) + W$ , and the canonical projection is $π (v) : = v + W$ (The quotient vector space $V / W$ and its canonical projection).

Proof

technique · direct

1.1L1L2L3

Define $\overset{ˉ}{f} (v + W) : = f (v)$ ; if $v + W = v^{'} + W$ , then $v - v^{'} \in W \subseteq \ker f$ by [L1], so $f (v) - f (v^{'}) = f (v - v^{'}) = 0$ by [L2] and the linearity of $f$ , giving $f (v) = f (v^{'})$ , and $\overset{ˉ}{f}$ is well defined; the operation formulas of [L3] then give $\overset{ˉ}{f} (a (v + W) + b (u + W)) = \overset{ˉ}{f} ((a v + b u) + W) = f (a v + b u) = a f (v) + b f (u)$ , so $\overset{ˉ}{f}$ is linear.

2.1step 1.1L1∎

The definition gives $\overset{ˉ}{f} (π (v)) = f (v)$ , hence $\overset{ˉ}{f} π = f$ ; if $g : V / W \to U$ also satisfies $g π = f$ , then every coset is $π (v)$ and $g (v + W) = f (v) = \overset{ˉ}{f} (v + W)$ , so $g = \overset{ˉ}{f}$ , including the cases $W = V$ and $V = 0$ .

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Cornell Math 4330, Quotient Spaces, Theorem 2 (standard reference, not scraped)