Matrix Subspaces: Does Set W = {X: AX=2X} Form a Subspace of M(2,1)?

[a.merchant]

Homework Statement

Let A be a fixed 2x2 matrix. Assuming that the set:
W={X:AX=2X}
has infinitly many solutions, determine whether it is a subspace of M(2,1)

Homework Equations

To determine whether a set is a subspace i need to prove that there is a zero vector, that it is closed under addition and scalar multiplication

The Attempt at a Solution

If A is 2x2 then AX can't be a subspace of a 2x1 space can it? Unless X is 2x1 but without knowing what X is how to I begin to prove that it is a subspace? Please help!

 

[HallsofIvy]

The wording of the problem is a bit confusing. I think you mean that W is the set of all 2 by 1 matrices (2 component column vectors), X, such that AX= 2X has infinitely many solutions. But since A is fixed, you are really just defining W as the set of all solutions of AX= 2X and asking when that has infinitely many solutions- that depends on A, not X.

As for your question about AX being in M(2,1), you are misreading the question. It does not say AX is in M(2,1), it is asking about the set of all X in M(2,1) such that AX= 2X.

I think what you are really asking is this: suppose A is such that AX= 2X has infinitely many solutions. Show that W, the set of all such solutions, is a subspace of M(2,1).

To do that, show the standard 3 things:
1) The set is non-empty. (Typically, show that the 0 matrix is in the set.)
2) The set is closed under addition: If AX= 2X and AY= 2Y then A(X+ Y)= 2(X+ Y).
3) The set is closed under scalar multiplication: If AX= 2X and k is any scalar, then A(kX)= 2(KX).

 

[a.merchant]

So i could just state that:

0 E W as the zero matrix times any 2x2 matrix will be all 0's, so it is equal to 2x the zero matrix (not in so many words, I am just not sure how to write matrices in this forum)

for the zero part?