Solution x spanning columns of A

[fackert]

This is a really odd question to me. Usually we talk about b (in Ax=b) spanning the columns of A but here its talking about x spanning A.

If x=<-3,4,8> (this is a vertical 3x1 matrix) is a solution to Ax=0, then is x in the set spanned by the columns of A?

I'm pretty sure it is no, but i can't explain why? (briefly). And i don't know how to change the statement so it would be true.

 

[HallsofIvy]

One obvious point- if A is an m by n matrix (m rows, n columns), then A maps an Rⁿ to R^m. In order that we be able to multiply A times x, x must have n rows and so must be in Rⁿ. But the columns each have m elements, one for each of the n rows, and so span a subspace of R^m. Members of the "column space" are vectors in the image of A, vectors of the form y= Ax for some x, not x itself.

 

[halo31]

Actually if its a solution to the matrix equation Ax=0 that's considered a nullspace.

 

[HallsofIvy]

Thanks, Halo31. fackert, if A maps vector space U to a subspace of vector Space V, then all solutions of Ax= 0, the "null space" of A, are in U. The columns of A, thought of as vectors (the "column space" of A), are in V. Of course, any x that A can be applied to must be in U, not V.