# Thread: Subspace problem

1. ## Subspace problem

Hi, I've been trying to figure out this problem, but I'm not quite sure how to go about it.

Problem:
For any matrix A(m,n) show that the set of right hand sides b in R^m for which Ax = b is solvable is a subspace of R^m

Attempt at a solution:
I know how to prove the solvability of a specific linear system, but how can I show that is true for any matrix? (Assuming I need to)

Any help/advice will be much appreciated!

2. Let $\displaystyle S\subset\mathbb{R}^m$ be the set of all b's which make $\displaystyle Ax=b$ solvable.

Consider that for an arbitrary $\displaystyle x1$ and $\displaystyle x2$ we have $\displaystyle A(x1) = b1$ and $\displaystyle A(x2) = b2$.

If we take the sum $\displaystyle A(x1)+A(x2)=b1+b2 \Rightarrow A(x1+x2)=(b1+b2) \Leftrightarrow (b1+b2) \in S$.

In the same way, we can have, for $\displaystyle k\in\mathbb{R}$, $\displaystyle kA(x1) = kb1 \Rightarrow A(kx1)=kb1 \Leftrightarrow kb1 \in S$.

I hope this is what you're looking for.

3. Yes this helps a lot! Thanks!

4. Originally Posted by MEMPHIS
Hi, I've been trying to figure out this problem, but I'm not quite sure how to go about it.

Problem:
For any matrix A(m,n) show that the set of right hand sides b in R^m for which Ax = b is solvable is a subspace of R^m

Attempt at a solution:
I know how to prove the solvability of a specific linear system, but how can I show that is true for any matrix? (Assuming I need to)

Any help/advice will be much appreciated!
You're given A is mxn.
You might give some consideration to the column space of A, which we (should) know is a subspace of R^m.

Can such a b fail to reside in the column space of A?
Is there a vector in the column space of A, say b, where we find that there is no solution set in R^n for the system Ax=b?