letAbe 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

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- Apr 15th 2010, 09:58 AMbugal402Linear 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 PMDrexel28
- Apr 16th 2010, 03:59 AMHallsofIvy
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 a**u**+ b**v**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 A**x**=**u**and A**y**=**v**.

Look at A(a**u**+ b**v**).