Math 2.0

statement theorem

A space is the direct sum of a subspace and its orthogonal complement

Statement

Let VV be an inner product space over R\R or C\C, and let WW be a finite-dimensional subspace of VV. Then V=W⊕W⊥,V = W \oplus W^\perp, where W⊥W^\perp is the orthogonal complement of WW and ⊕\oplus is the direct sum.

Remarks

For an orthonormal basis e1,…,eke_1, \dots, e_k of WW, each v∈Vv \in V is w+(v−w)w + (v - w) with w=∑i⟨v,ei⟩ei∈Ww = \sum_i \langle v, e_i \rangle e_i \in W and v−w∈W⊥v - w \in W^\perp. The statement holds for each closed subspace of a Hilbert space. It can fail for a subspace that is not closed in an infinite-dimensional space.

Explanations

L1 human

Each vector splits in exactly one way into a part in WW and a part perpendicular to WW. The part in WW is the shadow of the vector on WW. The part perpendicular to WW is what remains.

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionA space is the direct sum of a subspace and its orthogonal complement, statement, no evidence levelA space is the direct…statement
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  1. Vector space definition
  2. Inner product definition
  3. Linear subspace definition
  4. Orthogonal vectors definition
  5. Sum of subspaces definition
  6. Orthogonal complement definition
  7. Direct sum of subspaces definition
  8. A space is the direct sum of a subspace and its orthogonal complement statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Orthogonal_complement, Properties

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