Question

Say I have two different basis, do they span the same column space. being more explicit, if say i have vector (1,1,1) (1,2,3) that make one basis and (2,4,10), (1,1,3) makes another basis. Do they span the same column space?

Answer 1

It's possible. Bases are not unique. For example, consider R^2. You can define one basis as the xy coordinate grid that you see in high school (or in linear algebra terms, (1,0) and (0,1)). Then, you can also define a basis in polar coordinates. Thus, you have two different bases but they both span all of R^2.

Answer 2

It is possible but not necessarily. I think one of the real tests is to combine the two bases into a matrix to see whether the rank of this matrix is larger than the dimension of either one of the previous basis. (Assuming you are talking about two bases having the same dimension.