Linear and multilinear algebra; matrix theory Theorem

Universal property of the quotient space

Statement

Let $F$ be a field, let $V$ and $U$ be vector spaces over $F$ , and let $W$ be a subspace of $V$ . Let $\pi \colon V \to V / W$ be the quotient map. Let $T \colon V \to U$ be a linear map with $W \subseteq \ker T$ . Then there is exactly one linear map $\bar T \colon V / W \to U$ with $\bar T \circ \pi = T$ . It is given by $\bar T(v + W) = T(v)$ .

Remarks

The condition $W \subseteq \ker T$ makes $\bar T$ well defined. The kernel of $\bar T$ is $\ker T / W$ , and its image is $\im T$ . For $W = \ker T$ , the map $\bar T$ is injective, which gives the first isomorphism theorem.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Vector space definition
Linear map definition
Linear subspace definition
Quotient space definition
Quotient map definition
Universal property of the quotient space statement

Tag	`0ZERJX6PH6`
URI	`m2://0ZERJX6PH6`
Kind	statement, theorem
Writing	W0 stub
MSC	15A04
Status	This copy of the site has no build data. Run `m2 status 0ZERJX6PH6`.
Uses	Quotient map, Linear map

Universal property of the quotient space

Statement

Remarks

Reading path

Further reading

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