Let S be the subspace of R4 containing all vectors with x1 + x2 + x2 + x4 = 0. Find a basis for the space S orthogonal, containing all vectors orthogonal to S.

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- Oct 24th 2010, 12:35 PMveronicak5678Orthogonal Subspaces
Let S be the subspace of R4 containing all vectors with x1 + x2 + x2 + x4 = 0. Find a basis for the space S orthogonal, containing all vectors orthogonal to S.

- Oct 24th 2010, 12:45 PMTheEmptySet
There are a few ways to solve this but first note that basis for S is

So you could just find a vector perpendicular to these three above vectors.

Or if you take the gradient of you get

you can check using the vectors above that this is orthogonal to the space S. - Oct 24th 2010, 01:00 PMveronicak5678
I don't know about the gradient, but could you tell me how to find a vector that is perpendicular to those three?

- Oct 24th 2010, 02:20 PMzzzoak
You have three vectors

**a**,**b**and**c**.

Vector**p**is perpendicular to**a**,**b**and**c**if

**ap**=0

**bp**=0

**cp**=0

You have 3 equations and 4 unknown in**p**.

So one coordinate in**p**you may choose. - Oct 24th 2010, 02:37 PMAlso sprach Zarathustra