Thread: Stuck on Subspace proof question

Question:

Let A and B be mxn matrices and let S be the set of all x in R^n such that Ax = Bx. Show that S is a subspace of R^n.

My approach:
So i know that there are 3 properties for H to be a subspace in R^n.
1) the zero vector is in H
2) for each u and v in H, the sum u+v is in H
3) for each u in H and each scalar c, the vector cu is in H

So im confused on how i would go abouts proving this?
Would i take property 3 and multiply Ax=BX by a scalar?

Originally Posted by kensington

Question:

Let A and B be mxn matrices and let S be the set of all x in R^n such that Ax = Bx. Show that S is a subspace of R^n.

My approach:
So i know that there are 3 properties for H to be a subspace in R^n.
1) the zero vector is in H
2) for each u and v in H, the sum u+v is in H
3) for each u in H and each scalar c, the vector cu is in H

So im confused on how i would go abouts proving this?
Would i take property 3 and multiply Ax=BX by a scalar?

Sounds good to me!
1) What are A0 and B0? Are they equal? (0 is the 0 vector)
2) Suppose x and y are such that Ax=Bx and Ay= By. What are A(x+y) and B(x+y)? Are they equal?
3) Suppose x is such that Ax= By andck is a scalar. What are A(cx) and B(cx)? Are they equal?