# Linear Algebra matrix problem

• Apr 15th 2010, 09:58 AM
bugal402
Linear Algebra matrix problem
let A be an m x n matrix. prove that the set b = (vertor b a member of R^m : A*vector x = vector b is consistant) is a subspace of R^m

Thanks
• Apr 15th 2010, 12:02 PM
Drexel28
Quote:

Originally Posted by bugal402
let A be an m x n matrix. prove that the set b = (vertor b a member of R^m : A*vector x = vector b is consistant) is a subspace of R^m

Thanks

I don't know what this means. Let $\displaystyle A$ be an $\displaystyle m\times n$ matrix. Prove that the set $\displaystyle b=\left\{\bold{b}\in\mathbb{R}^m:A*\bold{x}=\bold{ b}\text{ is consistent}\right\}$?
• Apr 16th 2010, 03:59 AM
HallsofIvy
I think you mean "the set of all b such that Ax= b for some x in $\displaystyle R^m$" - that is known as the image of $\displaystyle R^m$ and is also the "column space" since it is spanned by the columns of the matrix.

To prove any subset is a subspace, it is sufficient to show that it is closed under addition and scalar multiplication- or, combining the two, that if u and v are in the subset and a and b are scalars, then au+ bv is also in the subset.

If u and v are in the subset then there exist x and y in $\displaystyle R^m$ such that Ax= u and Ay= v.

Look at A(au+ bv).