A subspace must contain 0. The span of any other set of vectors consists of all linear combinations of those vectors. Linear dependence and independence of vectors.
Vectors are dependent if and only if they have a nonzero linear relation.
The notion of basis. Any two bases for V have the same number of elements. The dimension of the subspace V is the number of vectors in a basis for V, which does not depend on which basis one chooses. The columns are independent if and only if the rank of A is equal to the number of columns. C has a solution.