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mammothrob
04-30-2007, 01:28 AM
Prove that the orthogonal complement of a subspace of (Rn) is itself a subspace of (Rn)

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Let V be the orthogonal complement of S, S a subspace of (Rn).

Let the set of vectors that span (Rn) be written as the columns of matrix A.

consider the homogenous equation
A^T \overline u = \overline 0

The solution space of the vectors u will all dot with any row vector from A transpose equaling zero.

So the null space of A transpose is the subspace V.

By (Fundamental Subspace Theroem) Two subspaces, Column space of a matricies transpose and the nullspace of that same matrix form a direct sum of (Rn).

This V is also a subspace of (Rn)

Does this make sense?
Am I trying way too hard here becuase this seems like it should be an easy one.

JakeD
04-30-2007, 05:30 AM
....Does this make sense?
Am I trying way too hard here becuase this seems like it should be an easy one.
Yes. Just use the definition of orthogonal complement and show it is closed under addition and scalar multiplication.