# Subspaces and Basis

1. Nov 8, 2008

### tracedinair

1. The problem statement, all variables and given/known data

Let u be a vector where u = [4 3 1]. Let A be the set of all vectors orthogonal to u. Show that A is subspace of R^3. Then find the basis for A.

2. Relevant equations

3. The attempt at a solution

For showing that A is a subspace...

Zero vector is in A because A(0) = 0

For any u & v, u+v is in A because Au=0, Av=0, and A(u+v) = Au+Av = 0

And for any scalar c, A(cu) = c(Au) = c(o) = 0

As for the basis, I really have no idea where to even start with that.

Thanks for any help.

2. Nov 8, 2008

### Staff: Mentor

You should do the first part of this problem; namely, finding the set of vectors that are orthogonal to u = (4, 3, 1). How can you tell that an arbitrary vector (x, y, z) is orthogonal to a given vector?
None of this makes any sense. A is a set, not a matrix, so it doesn't make any sense to multiply a vector by A.

As for finding a basis for A, if you do the first part you will be on your way toward a basis.

3. Nov 9, 2008

### tracedinair

I'm not entirely sure how to show an arbitrary vector is orthogonal to a given vector. I've looked through my text for help, but it's not really helping.

4. Nov 9, 2008

### Staff: Mentor

Is "dot product" a hint?