1. **Problem Statement:** Understand the concepts of orthogonal and orthonormal vectors, orthogonal and orthonormal bases, orthogonal matrices, and their properties in linear algebra. 2. **Orthogonal Vectors:** Two vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^n$ are orthogonal if their dot product is zero: $$\mathbf{u} \cdot \mathbf{v} = 0.$$ This means they are perpendicular to each other. 3. **Orthogonal Set:** A set of vectors $\{\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_p\}$ is orthogonal if every pair of distinct vectors in the set is orthogonal, i.e., $$\mathbf{u}_i \cdot \mathbf{u}_j = 0 \text{ for } i \neq j.$$ This ensures all vectors are mutually perpendicular. 4. **Orthonormal Set:** An orthonormal set is an orthogonal set where each vector is a unit vector (length 1). Formally, $$\mathbf{u}_i \cdot \mathbf{u}_i = 1 \text{ for all } i.$$ This means vectors are perpendicular and normalized. 5. **Orthogonal Basis:** A basis for a subspace $W$ of $\mathbb{R}^n$ is orthogonal if the basis vectors form an orthogonal set. This simplifies many computations like projections. 6. **Orthonormal Basis:** An orthonormal basis is an orthogonal basis where all basis vectors are unit vectors. This is very useful because it preserves lengths and angles. 7. **Orthogonal Matrix:** A square matrix $P$ is orthogonal if its inverse equals its transpose: $$P^{-1} = P^T,$$ which implies $$P^T P = I,$$ where $I$ is the identity matrix. The rows and columns of $P$ form orthonormal sets. 8. **Examples:** - Vectors $\mathbf{u} = (1,2)$ and $\mathbf{v} = (6,-3)$ are orthogonal since $$1 \times 6 + 2 \times (-3) = 6 - 6 = 0.$$ - The set $\{(3,1,1), (-1,2,1), (-\frac{1}{2}, -2, \frac{7}{2})\}$ is orthogonal because all pairwise dot products are zero. - The set $\{(\frac{\sqrt{3}}{11}, \frac{\sqrt{1}}{11}, \frac{\sqrt{1}}{11}), (-\frac{\sqrt{1}}{6}, \frac{\sqrt{2}}{6}, \frac{\sqrt{1}}{6}), (-\frac{\sqrt{1}}{66}, -\frac{\sqrt{4}}{66}, \frac{\sqrt{7}}{66})\}$ is orthonormal because vectors are orthogonal and each has length 1. - The matrix $$P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$$ is orthogonal since $$P^{-1} = P^T.$$ 9. **Summary:** Orthogonal and orthonormal vectors simplify many linear algebra operations. Orthogonal matrices preserve vector lengths and angles, making them important in transformations and decompositions like QR factorization and singular value decomposition.