The four fundamental spaces

• Null space: $N(A)=\{x|Ax=0\}$.
• Column space or range: $R(A)=\{y|y=Ax\}$.
• Row space: $R(A^T)$.
• Null space of $A^T$: $N(A^T)$.

The row space and column space have the same dimension. It’s called the rank $r$ of the matrix.

Rank-nullity theorem: The dimension of the null space is related to the dimension of the range:

$$\text{dim}(R(A))+\text{dim}(N(A))=n$$

Orthogonal complement theorem: $R(A{)}^{\perp }=N(A^T)$ (and similarly for $A^T$).

Subspace and linear independence, Vectors and matrices