#Orthogonal Vectors
Two vectors are orthogonal if angle between vectors is 90 degrees.
#Example
#Exercise
- Find a vector $\mathbf{u}$ (no zero vector) where $\text{Angle} (\mathbf{u}, \mathbf{v}) = 90^\circ$ and $\mathbf{v} = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}$.
#Orthogonal Subspaces
Subspace S is orthogonal to subspace T ($T = S^\perp$), means every vector in S is orthogonal in T.
#Orthogonal Matrix
A matrix orthogonal with Orthonormal columns is assigned special letter Q.
$$ Q^T Q = \begin{pmatrix} \vdots & \mathbf{q}_1^T & \vdots \\ \vdots & \mathbf{q}_n^T & \vdots \end{pmatrix} \begin{pmatrix} \vert & & \vert \\ \mathbf{q}_1 & \cdots & \mathbf{q}_n \\ \vert & & \vert \end{pmatrix} = I $$If Q is square matrix $Q^T Q = I$ tells us that $Q^T=Q^{-1}$
#Example
Null Space