Orthogonality and orthogonal Subspaces

Two subspaces are orthogonal when all vectors $v$ and $w$ in them satisfy the fundamental condition of orthogonality.

$$v^{T}w=w^{T}v=0$$

Conditions

• Orthogonality is impossible when the dimension of the two compared spaces is larger than the dimension of the space they both are in

Orthogonality of the four fundamental Subspaces

• Row Space $\perp$ Nullspace by definition of $Ax=0$. The spaces only meet with $\ne 0°$ at $Z$. $N(A)$ is the Orthogonal Complement of $C(A^T)$.
• Column Space $\perp$ Left Null Space In the exact same manner.

Orthogonality of two planes in $R^3$

While the normal vectors $\vec{n}$ of two planes $R^2$ (Two walls meeting in the corner of a 3D room for example) must are orthogonal, their subspaces are not because they contain vectors that satisfy $v\_{p\_1}^{T}v\_p\_2\ne 0$. Just don’t confuse the normal vector $\vec{n}$ with the subspace vectors $v$ themselves.