Math 2.0

statement corollary

Dimension of the orthogonal complement

Statement

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

Remarks

It follows from V=W⊕W⊥V = W \oplus W^\perp. For example, in R3\R^3, a line has a plane as orthogonal complement: 1+2=31 + 2 = 3. For a real matrix, it gives rank⁡A+dim⁡ker⁡A=n\rank A + \dim \ker A = n once more, because ker⁡A\ker A is the orthogonal complement of the row space.

Explanations

L1 human

A subspace and its orthogonal complement share the dimensions of the space. If WW uses kk directions, then W⊥W^\perp uses the remaining n−kn - k directions.

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionDimension of the orthogonal complement, statement, no evidence levelDimension of the orth…statement
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  1. Vector space definition
  2. Inner product definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Orthogonal vectors definition
  6. Linear independence definition
  7. Linear span definition
  8. Orthogonal complement definition
  9. Spanning set definition
  10. Basis definition
  11. Finite-dimensional vector space definition
  12. Dimension definition
  13. Dimension of the orthogonal complement statement

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Sources

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

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