Subspace & Basis: Proving A is a Subspace of R^3

[tracedinair]

Homework Statement

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.

Homework Equations

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.

 

[Mark44]

Homework Statement

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.

Homework Equations

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.

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?

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

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.

 

[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.

 

[Mark44]

Is "dot product" a hint?