let V be a finite dimentional inner product space and let W be a subspace of V. prove that dim(W) + dim(W^)= dim(V) where W^ is a set of vectors orthogonal to W.

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- January 8th 2009, 03:12 PMKat-Mlinear algebra problem
let V be a finite dimentional inner product space and let W be a subspace of V. prove that dim(W) + dim(W^)= dim(V) where W^ is a set of vectors orthogonal to W.

please help. - January 8th 2009, 05:23 PMHallsofIvy
Since W is a subspace of an innerproduct space, we can select an orthonormal basis for W containing dim(W) vectors. Since W^ is a subspace, we can select an orthonomal basis for W^ containing dim(W^) vectors. Show that their union is a basis for the entire space.