Math 2.0

statement theorem

Existence of a complementary subspace

Statement

Let FF be a field, let VV be a vector space over FF, and let UU be a subspace of VV. Then UU has a complement in VV: there is a subspace WW of VV with V=U⊕WV = U \oplus W.

Remarks

Proof: extend a basis BUB_U of UU to a basis BB of VV, and let WW be the span of B∖BUB \setminus B_U. For infinite-dimensional VV, this uses the extension of independent sets to bases, and hence Zorn's lemma. The statement fails for modules over rings: the submodule 2Z2\Z of Z\Z has no complement. For closed subspaces of topological vector spaces, a closed complement need not exist.

Explanations

L1 human

Every subspace has a partner subspace that fills up the rest of the space. We find it by completing a basis of the subspace and taking the span of the new vectors. The partner is not unique, but its dimension is fixed.

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Vector space, definition, no evidence levelVector spacedefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionComplementary subspaces, definition, no evidence levelComplementary subspac…definitionExistence of a complementary subspace, statement, no evidence levelExistence of a comple…statement
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  1. Vector space definition
  2. Linear subspace definition
  3. Sum of subspaces definition
  4. Direct sum of subspaces definition
  5. Complementary subspaces definition
  6. Existence of a complementary subspace statement

Sources

  1. Web https://en.wikipedia.org/wiki/Complemented_subspace, Algebraic direct sum

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