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Math Help - question about dimension

  1. #1
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    question about dimension

    to prove that given a vector space U and its subspace W, dim U = dim W + dim (orthogonal complement of W), would this proof work?

    suppose {w1,...,wk} is an orthonormal basis for W.
    assuming that dim U = n, we can extend this basis to {w1,...,wk,w_k+1,...,wn} which is a basis for U.
    then {w_k+1,...,wn} must be a basis for the orthogonal complement since W is in the span of {w1,...,wk}.
    so our claim follows.
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  2. #2
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    Quote Originally Posted by squarerootof2 View Post
    to prove that given a vector space U and its subspace W, dim U = dim W + dim (orthogonal complement of W), would this proof work?

    suppose {w1,...,wk} is an orthonormal basis for W.
    assuming that dim U = n, we can extend this basis to {w1,...,wk,w_k+1,...,wn} which is a basis for U.
    then {w_k+1,...,wn} must be a basis for the orthogonal complement since W is in the span of {w1,...,wk}.
    so our claim follows.
    That is more or less correct. But you need to say that the orthonormal basis for W can be extended to an orthonormal basis for U. (This is possible, for example by using the Gram–Schmidt process.)
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