Question

We define
A = 1 1 1 B = 1 2 3
2 2 2 1 2 3
3 3 3 1 2 3
Suppose W = { x | Ax = Bx }. Is W a vector space? If it is, find the linear basis for W.

If I am not mistaken then the only members of W is the scalar 0 and the 3x3 matrix whose entries are all zero.
How can I mathematically prove whether or not that the only members in W are these 'zeroes'?

Answer 1

when you are trying to prove that something is a vector space, you do really care what is in the space, so dont try to find whats in it.

You are only concerned with closure under addition, and closure under scalar multiplication. So you have to ask yourself, if i have two vectors, say x and y, that are in W, if I add them, is the resulting vector still in W? If i have a vector x and a scalar c, and i do scalar vector multiplication and get cx, is the new vector cx still in W?

If you can answer those two questions (with proof), then you can say for sure a space is a vector space, even without knowing what is in the space exactly.

Answer 2

you dont* really care....my bad.

Answer 3

Why is it I don't have to worry about meeting the criteria for things like Commutativity, Association, Distribution, Identity, Inverse, and all that good stuff?

Answer 4

Im running under the assumption that we are using Real Numbers here. Because Reals are a Field, we are granted those properties. The only time you need to verify those aspects is when you arent sure what properties the elements in your vectors/matrices have.

Answer 5

unless your professor is mean. then by all mean you would have to prove that lolol

Answer 6

Oh I think I understand... so you're saying if we can show that it is a subspace of R by showing closure under the two binary operations then we can assert that it is a vector space, because subspaces are vector spaces?

Answer 7

yep, that is correct.

Answer 8

Nice, thanks a million Joe

Answer 9

np :)