Orthogonal Vectors

Two vectors are orthogonal (perpendicular) if their dot product is zero:

$$\mathbf{u} \cdot \mathbf{v} = 0$$

Pythagorean Theorem

For orthogonal vectors:

$$\|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2$$

Orthogonal Subspaces

Two subspaces are orthogonal if every vector in one is orthogonal to every vector in the other.

• $C(A) \perp N(A^T)$ — column space is orthogonal to left null space
• $C(A^T) \perp N(A)$ — row space is orthogonal to null space