I know what is the column space of a matrix: it is basically the subspace formed by the linear combinations of the columns (vectors) of a matrix. From wikipedia, we have the following nice picture:
Which shows up the vectors that are used to make any linear combination to form the column space.
So, my problem is not strictly related to the concept of column space of a matrix, but why is it useful?
Before knowing the concept of column space, I already was used to find, for example, the RREF of a matrix, to see if the columns for a basis or are linearly independent or whatever... What's in the connection between column space and all those things?