Null space and column space

For a matrix $A$, the vector space spanned by the solutions $\{\mathbf x\}$ to the linear equation $A \mathbf x = \mathbf 0$ is called the null space $\mathrm{NS}(A)$.

The dimensions of the null space $\dim(\mathrm{NS}(A))$ is the number of non-pivot columns of the row echelon form $\mathrm{REF}(A)$. It is also the number of free variables or the “nullity” of $A$.

The column space $\mathrm{CS}(A)$ is the vector space spanned by the columns of $A$. To find the basis of the column space we can find $B = \mathrm{REF}(A)$ and locate the pivots of $B$. The corresponding columns in $A$ form the basis of the column space. It is important to note that $\mathrm{CS}(A) \ne \mathrm{CS}(\mathrm{REF}(A))$.