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.

-----------------------------------------------------

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.