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.

 

[blue_raver22]

I would first clarify the terminology used in this question. When we say "x spanning columns of A", it typically means that the vector x is a linear combination of the columns of A. In other words, x can be written as a linear combination of the columns of A, with the coefficients of the linear combination being the entries of x.

In this context, it is not correct to say that x is spanning the columns of A. Rather, the columns of A are spanning the vector space that x belongs to. Therefore, the statement "x spanning A" is not accurate.

Moving on to the question at hand, if x is a solution to Ax=0, then it means that x is in the null space of A. The null space of A is the set of all vectors that when multiplied by A, result in the zero vector. This means that x is orthogonal to all the columns of A.

Since x is orthogonal to the columns of A, it cannot be written as a linear combination of the columns of A. Therefore, x is not in the set spanned by the columns of A.

To change the statement to make it true, we could say "If x is a solution to Ax=0, then x is in the null space of A." This statement accurately reflects the relationship between x and the columns of A.