Inner Product, Length and Orthogonality
36 min 10 Examples
- Overview of the Inner Product, Length, and Orthogonality
- Finding Inner Product and Length for given vectors (Examples #1-4)
- Distance between two vectors and Orthogonal Vectors (Examples #5-6)
- Four Examples – Determining if given vectors are Orthogonal (Examples #7-8)
- Orthogonal Complements and their applications (Examples #9-10)
Orthogonal Sets
35 min 8 Examples
- Orthogonal Set and Orthogonal Basis Theorem (Example #1)
- Expressing a vector as a Linear Combination of the Orthogonal Basis (Example #2)
- Orthogonal Projections and computations (Examples #3-5)
- Orthonormal Sets and determining if sets are Orthonormal (Examples #6-8)
Orthogonal Projections and Decomposition
31 min 6 Examples
- Orthogonal Projections, Decomposition Theorem, and Best Approximation Theorem
- Finding Orthogonal Projection and verifying Orthogonal Sets (Examples #1-2)
- Writing y as the sum of vectors and finding the closest point to y (Examples #3-5)
- Matrix Multiplication insights (Example #6)
The Gram-Schmidt Process and QR Factorization
47 min 5 Examples
- Gram-Schmidt Process, creating Orthogonal Basis, and Orthonormal Basis (Examples #1-3)
- QR Factorization overview and computations (Examples #4-5)
Least-Squares Problems
28 min 4 Examples
- Least-Squares Problem, Theorem, Methods, and Least-Squares Error (Example #1)
- Finding the Least-Squares Solution (Examples #2-3)
- Alternative Calculations of Least-Squares Solutions (Example #4)
Applications to Linear Models: Least-Squares Lines
29 min 4 Examples
- Least-Squares Lines, Residuals, Predicted and Observed Values (Example #1)
- Finding the Least-Squares Line and Linear Regression Model (Examples #2-3)
- Computing Least-Squares Curves (Example #4)
Inner Product Spaces
54 min 9 Examples
- Inner Product Spaces, Axions, and Lengths
- Distances, Orthogonality, and Polynomial Inner Products (Examples #1-2)
- Matrix-generated Inner Products and Best Approximation Theorem (Example #3-4)
- Calculus of Inner Product Spaces (Examples #5-8)
- Cauchy-Schwarz Inequality and Triangle Inequality (Example #9)
Chapter Test
1 hr 22 min 15 Problems
- Dot products, unit vectors, and distances (Problems #1-3)
- Orthogonal sets and projections (Problems #4-6)
- Closest points, inner products, and orthonormal basis (Problems #7-11)
- QR factorization, least-squares solutions, and errors (Problems #12-14)
- Least-squares lines for data points (Problem #15)