mammothrob
Junior Member
- Joined
- Nov 12, 2005
- Messages
- 91
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
\(\displaystyle A^T \overline u = \overline 0\)
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
\(\displaystyle A^T \overline u = \overline 0\)
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.