Linear and multilinear algebra; matrix theory Linear maps Theorem

First isomorphism theorem for linear maps

Builds on Quotient space Kernel of a linear map Image of a linear map

Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$ , and let $T \colon V \to W$ be a linear map. Then the map $\bar T \colon V / \ker T \to \im T, \qquad v + \ker T \mapsto T(v),$ is well defined, and it is a linear isomorphism. Here $V / \ker T$ is the quotient space by the kernel, and $\im T$ is the image of $T$ .

Remarks

In finite dimensions, it gives the rank–nullity theorem: $\dim V - \dim \ker T = \dim \im T$ . The same theorem holds for groups, rings and modules. For a surjective map $T$ , it says $V / \ker T \cong W$ .

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Vector space definition
Linear map definition
Linear subspace definition
Image of a linear map definition
Kernel of a linear map definition
Quotient space definition
First isomorphism theorem for linear maps statement

Tag	`0MTBGVZ4XJ`
URI	`m2://0MTBGVZ4XJ`
Kind	statement, theorem
Writing	W0 stub
MSC	15A04, 16D10
Status	This copy of the site has no build data. Run `m2 status 0MTBGVZ4XJ`.
Uses	Quotient space, Kernel of a linear map, Image of a linear map

First isomorphism theorem for linear maps

Statement

Remarks

Reading path

Further reading

What this page needs for W1

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