Math 2.0

statement proposition

Orthogonal complement 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 (W⊥)⊥=W,(W^\perp)^\perp = W, where W⊥W^\perp is the orthogonal complement of WW.

Remarks

Each vector of the subspace is orthogonal to each vector of its orthogonal complement, so W⊆(W⊥)⊥W \subseteq (W^\perp)^\perp. Both have dimension dim⁡V−dim⁡W⊥=dim⁡W\dim V - \dim W^\perp = \dim W. In an infinite-dimensional Hilbert space, (W⊥)⊥(W^\perp)^\perp is the closure of WW.

Explanations

L1 human

Taking the orthogonal complement twice gives back the original subspace. The complement of the complement of a line is the same line.

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To read this node, read these 14 nodes in this order. The foundations come first, and the list ends with this node.

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…statementOrthogonal complement of the orthogonal complement, statement, no evidence levelOrthogonal complement…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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
  14. Orthogonal complement of the orthogonal complement statement

Sources

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

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