Proving Orthogonal Compliments of Subspaces in Matrix Algebra

[mlarson9000]

Homework Statement

Let A be an mxn matrix.
a. Prove that the set W of row vectors x in R^m such that xA=0 is a subspace of R^m.

b. Prove that the subspace W in part a. and the column space of A are orthogonal compliments.

Homework Equations

The Attempt at a Solution

a. to be a subspace, I believe i only need to show that W is closed under addition and multiplication. So I just show that (rx+sy)A=0. Right?

b. Not too sure about this. Should I try to show that x dotted with a=0 for all x and a? Or should try to do something with the properties of orthogonal compliments? I can show that dim(W)= nullity(A), but I don't think that's really going to do anything for me.

 

[Office_Shredder]

For part a you have the right idea.

For part b, If you look at the row vector x dotted with one of the columns of W, say a, then where in the vector xA can you find that number?

 

[mlarson9000]

For a row vector x dotted with the column vector ai, the resuting value will be the ith column of the 1xn zero vector. This for me does not guarantee that the value will be zero, however.

 

[Dick]

For a row vector x dotted with the column vector ai, the resuting value will be the ith column of the 1xn zero vector. This for me does not guarantee that the value will be zero, however.

I don't get it. The result is the ith entry in a zero vector. How might that not be zero?

 

[mlarson9000]

If addition or scalar multiplication are redefined, then the zero vector can have nonzero entries. So since the problem doesn't say anything about that, I am to assume that everything is normal?

 

[Dick]

If they don't explicitly tell you to use a nonstandard addition or scalar product, then sure, assume everything is normal.