Math 2.0

statement theorem

Third isomorphism theorem for vector spaces

Statement

Let FF be a field, let VV be a vector space over FF, and let W⊆UW \subseteq U be subspaces of VV. Then U/WU / W is a subspace of V/WV / W, and the map (V/W)/(U/W)→V/U,(v+W)+U/W↦v+U,(V / W) / (U / W) \to V / U, \qquad (v + W) + U/W \mapsto v + U, is well defined, and it is a linear isomorphism of quotient spaces.

Remarks

The proof applies the first isomorphism theorem to the surjective map V/W→V/UV / W \to V / U, v+W↦v+Uv + W \mapsto v + U, whose kernel is U/WU / W. In finite dimensions, both sides have dimension dim⁡V−dim⁡U\dim V - \dim U.

Explanations

L1 human

Dividing first by a small subspace and then by a larger one is the same as dividing by the larger one at once. This works like cancelling in fractions. The same theorem holds for groups, rings and modules.

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Vector space, definition, no evidence levelVector spacedefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionQuotient space, definition, no evidence levelQuotient spacedefinitionThird isomorphism theorem for vector spaces, statement, no evidence levelThird isomorphism the…statement
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Sources

  1. Web https://en.wikipedia.org/wiki/Isomorphism_theorems, Theorem C (modules)

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