Thread: 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

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 be an matrix. Prove that the set ?

I think you mean "the set of all b such that Ax= b for some x in " - that is known as the image of 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 such that Ax= u and Ay= v.

Look at A(au+ bv).