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, 11:35 AMveronicak5678Orthogonal 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, 11:45 AMTheEmptySet
There are a few ways to solve this but first note that basis for S is

$\displaystyle (x_1,x_2,x_3,-x_1-x_2-x_3)=x_1(1,0,0,-1)+x_2(0,1,0,-1)+x_3(0,0,1,-1)$

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

Or if you take the gradient of $\displaystyle x_1+x_2+x_3+x_4=0$ you get

$\displaystyle (1,1,1,1)$ you can check using the vectors above that this is orthogonal to the space S. - Oct 24th 2010, 12:00 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, 01:20 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, 01:37 PMAlso sprach Zarathustra