Question

Could someone explain the difference between subspace and column space?

Answer 1

A subspace is a vector space contained by some larger vector space. In particular it has to have the zero vector within it. For Ax, the column space is the totality of all linear combinations of the columns of A with the elements of x being the coefficients of their respective columns in said combinations. Thus for Ax = b, b is within the column space of A. For a non-zero x, can b = 0? That is the question ....

Answer 2

Correction : 'contained by some larger vector space'. It's not necessarily larger. A vector space is a subspace of itself. There may well be vectors in one space not contained in the other, in which case 'larger' is true but that's not required in the definition of subspace.