Two vectors are orthogonal when their dot product is zero: $v · w = v ^T w = 0(||v||^2+||w||^2=||v+w||^2).$

The row space is perpendicular to the nullspace

Every row of $A$ is perpendicular to every solution of $Ax = 0.$

The nullspace $N(A)$ and the row space $C(A^T )$ are orthogonal subspaces of $R^n$ .

second proof : $x^T(A^Ty)=(Ax)^Ty=0^Ty=0$

The column space is perpendicular to the nullspace of $A^T$

Every vector y in the nullspace of $A^T$ is perpendicular to every column of $A$.

DEFINITION :

Two subspaces $V$ and $W$ of a vector space are orthogonal if every vector $v$ in $V$ is perpendicular to every vector $w$ in $W$:

$v ^T w = 0$ for all $v$ in $V$ and all $w$ in $W$.

Orthogonal Complements

DEFINITION:

The orthogonal complement of a subspace $V$ contains every vector that is perpendicular to $V$. This orthogonal subspace is denoted by $V^\bot$ (pronounced "$V$ perp").