1. ## Orthogonal 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.

2. There are a few ways to solve this but first note that basis for S is

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

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

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

$(1,1,1,1)$ you can check using the vectors above that this is orthogonal to the space S.

3. I don't know about the gradient, but could you tell me how to find a vector that is perpendicular to those three?

4. 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.

5. Originally Posted by veronicak5678
I don't know about the gradient, but could you tell me how to find a vector that is perpendicular to those three?
Look at x1 + x2 + x2 + x4 = 0 as a equation of plane in R^4, gradient is the perpendicular vector to that plane ( the coefficients of x1,x2,x3,x4, which is the vector: (1,1,1,1))