A subspace of a vector space is a set of vectors (including 0) that satisfies two requirements: If $v$ and $w$ are vectors in the subspace and $c$ is any scalar, then (1) $v + w$ is in the subspace (2) $cv$ is in the subspace.
A subspace containing v and w must contain all linear combinations $cv + dw$.
DEFINITION The column space consists of all linear combinations of the columns . The combinations are all possible vectors $Ax$. They fill the column space $C(A)$.
The system $Ax = b$ is solvable if and only if $b$ is in the column space of $A$.
The nullspace $N(A)$ consists of all solutions to $Ax= 0$. These vectors $x$ are in $R^n$
DEFINITION The columns of $A$ are linearly independent when the only solution to $Ax = 0$ is $x = 0$. No other combination $Ax$ of the columns gives the zero vector.
