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complementary subspace
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(Definition)
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Let $U$ be a vector space, and $V,W\subset U$ subspaces. We say that $V$ and $W$ span $U$ and write $$U=V+W$$ if every $u\in U$ can be expressed as a sum $$u=v+w$$ for some $v\in V$ and $w\in W$
If in addition, such a decomposition is unique for all $u\in U$ or equivalently if $$V\cap W=\{ 0\},$$ then we say that $V$ and $W$ form a direct sum decomposition of $U$ and write $$U=V\oplus W.$$ In such circumstances, we also say that $V$ and $W$ are complementary subspaces, and also say that $W$ is an algebraic complement of $V$
Here is useful characterization of complementary subspaces if $U$ is finite-dimensional.
Proposition 1 Let $U, V, W$ be as above, and suppose that $U$ is finite-dimensional. The subspaces $V$ and $W$ are complementary if and only if for every basis $v_1,\ldots, v_m$ of $V$ and every basis $w_1,\ldots,w_n$ of $W$ the combined list $$v_1,\ldots,v_m,w_1,\ldots,w_n$$ is a basis of $U$
Remarks.
- Since every linearly independent subset of a vector space can be extended to a basis, every subspace has a complement, and the complement is necessarily unique.
- Also, direct sum decompositions of a vector space $U$ are in a one-to correspondence fashion with projections on $U$
Specializing somewhat, suppose that the ground field $\kf$ is either the real or complex numbers, and that $U$ is either an inner product space or a unitary space, i.e. $U$ comes equipped with a positive-definite inner product $$\langle,\rangle:U\times U\rightarrow \kf.$$ In such circumstances, for
every subspace $V\subset U$ we define the orthogonal complement of $V$ denoted by $V^\perp$ to be the subspace $$V^\perp = \{ u\in U: \langle v,u\rangle = 0,\text{ for all } v\in V\}.$$
Proposition 2 Suppose that $U$ is finite-dimensional and $V\subset U$ a subspace. Then, $V$ and its orthogonal complement $V^\perp$ determine a direct sum decomposition of $U$
Note: the Proposition is false if either the finite-dimensionality or the positive-definiteness assumptions are violated.
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"complementary subspace" is owned by rmilson. [ full author list (2) ]
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(view preamble | get metadata)
| Other names: |
algebraic complement |
| Also defines: |
complementary, direct sum, decomposition, orthogonal complement |
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Cross-references: proposition, inner product, unitary space, inner product space, complex numbers, real, ground field, projections, complement, subset, linearly independent, basis, finite-dimensional, characterization, addition, sum, span, subspaces, vector space
There are 123 references to this entry.
This is version 8 of complementary subspace, born on 2002-07-26, modified 2008-06-01.
Object id is 3209, canonical name is Complimentary.
Accessed 30636 times total.
Classification:
| AMS MSC: | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) |
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Pending Errata and Addenda
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