Thread: 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.

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.

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

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.

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))