**Orthogonality and Least Squares**

## Inner Product, Length and Orthogonality

36 min 10 Examples

- Overview of the Inner Product and Length
- Four Examples – find the Inner Product and Length for the given vectors
- Overview of how to find Distance between two vectors with Example
- Overview of Orthogonal Vectors and Law of Cosines
- Four Examples – Determine if the given vectors are Orthogonal
- Overview of Orthogonal Complements with Example
- Orthogonal Complement and the Row, Null and Column Spaces with Example

## Orthogonal Sets

35 min 8 Examples

- Overview of an Orthogonal Set with Example
- Orthogonal Basis Theorem
- Example – Express the vector as a Linear Combination of the Orthogonal Basis
- Overview of Orthogonal Projections
- Example #1 – Compute the Orthogonal Projection
- Example #2 – Compute the Component of y orthogonal to u
- Example #3 – Write y as the sum of vectors in the span and orthogonal to u
- Overview of Orthonormal Sets
- Three Examples – Determine if the sets are Orthonormal

## Orthogonal Projections and Decomposition

31 min 6 Examples

- Review of an Orthogonal Projections and The Orthogonal Decomposition Theorem
- Example – Find the Orthogonal Projection
- Example – Verify the Orthogonal Set
- Example – Write y as the sum of vectors
- The Best Approximation Theorem with Example
- Example – Find the closest point to y
- Another way to view Matrix Multiplication with Example

## The Gram-Schmidt Process and QR Factorization

47 min 5 Examples

- Overview of the Gram-Schmidt Process
- Example #1 – Describe the Orthogonal Basis using the Gram-Schmidt Process
- Example #2 – Find the Orthogonal Basis using the Gram-Schmidt Process
- Example #3 – Find an Orthonormal Basis using the Gram-Schmidt Process from previous example
- Overview of QR Factorization
- Example #1 – Find the QR Factorization for the given matrix
- Example #2 – Find the QR Factorization for the given matrix

## Least-Squares Problems

28 min 4 Examples

- Overview of the Least-Squares Problem
- Theorem and Methods for Finding the Least-Squares Solution
- Example #1 – Find the Least-Squares Solution
- Example #2 – Find the Least-Squares Solution
- Least-Squares Error with Example
- Alternative Calculations of Least-Squares Solutions

## Applications to Linear Models: Least-Squares Lines

29 min 4 Examples

- Overview of the Least-Squares Lines with Definitions and Theorems
- Definition of Residuals and Overview of Predicted Values and Observed Values
- Steps for Finding the Least-Squares Line
- Example #1 – Find the Equation of the Least-Squares Line
- Example #2 – Find the Equation of the Least-Squares Line
- Example #3 – Find the Least-Squares Linear Regression Model
- Steps for Computing Least-Squares Curves
- Example #4 – Find the Equation of the Least-Squares Curve

## Inner Product Spaces

54 min 9 Examples

- Overview of Inner Product Spaces and Axions
- Example #1 – Prove the vectors form an Inner Product Space
- Example #2 – Show why the vectors do not form an Inner Product Space
- Lengths, Distances and Orthogonality Overview
- Example – use the Inner Product to compute Length and Distance
- Polynomial Inner Products with Example
- Inner Products Generated by Matrices with Example
- Best Approximation Theorem in Inner Product Spaces
- Example – Compute the Orthogonal Projection onto a subspace spanned by polynomials
- The Calculus of Inner Product Spaces with example
- Example #2 – Calculus of Inner Products
- Example #3 – Calculus of Inner Products
- Example #4 – Calculus of Inner Products
- The Cauchy-Schwarz Inequality and the Triangle Inequality
- Example – find the Inner Product using the Cauchy-Schwarz Inequality