# Linear Algebra

via YouTube

## Syllabus

What's the big idea of Linear Algebra? **Course Intro**.

What is a Solution to a Linear System? **Intro**.

Visualizing Solutions to Linear Systems - - 2D & 3D Cases Geometrically.

Rewriting a Linear System using Matrix Notation.

Using Elementary Row Operations to Solve Systems of Linear Equations.

Using Elementary Row Operations to simplify a linear system.

Examples with 0, 1, and infinitely many solutions to linear systems.

Row Echelon Form and Reduced Row Echelon Form.

Back Substitution with infinitely many solutions.

What is a vector? Visualizing Vector Addition & Scalar Multiplication.

Introducing Linear Combinations & Span.

How to determine if one vector is in the span of other vectors?.

Matrix-Vector Multiplication and the equation Ax=b.

Matrix-Vector Multiplication Example.

Proving Algebraic Rules in Linear Algebra --- Ex: A(b+c) = Ab +Ac.

The Big Theorem, Part I.

Writing solutions to Ax=b in vector form.

Geometric View on Solutions to Ax=b and Ax=0..

Three nice properties of homogeneous systems of linear equations.

Linear Dependence and Independence - Geometrically.

Determining Linear Independence vs Linear Dependence.

Making a Math Concept Map | Ex: Linear Independence.

Transformations and Matrix Transformations.

Three examples of Matrix Transformations.

Linear Transformations.

Are Matrix Transformations and Linear Transformation the same? Part I.

Every vector is a linear combination of the same n simple vectors!.

Matrix Transformations are the same thing as Linear Transformations.

Finding the Matrix of a Linear Transformation.

One-to-one, Onto, and the Big Theorem Part II.

The motivation and definition of Matrix Multiplication.

Computing matrix multiplication.

Visualizing Composition of Linear Transformations **aka Matrix Multiplication**.

Elementary Matrices.

You can "invert" matrices to solve equations...sometimes!.

Finding inverses to 2x2 matrices is easy!.

Find the Inverse of a Matrix.

When does a matrix fail to be invertible? Also more "Big Theorem"..

Visualizing Invertible Transformations (plus why we need one-to-one).

Invertible Matrices correspond with Invertible Transformations **proof**.

Determinants - a "quick" computation to tell if a matrix is invertible.

Determinants can be computed along any row or column - choose the easiest!.

Vector Spaces | Definition & Examples.

The Vector Space of Polynomials: Span, Linear Independence, and Basis.

Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples.

The Span is a Subspace | Proof + Visualization.

The Null Space & Column Space of a Matrix | Algebraically & Geometrically.

The Basis of a Subspace.

Finding a Basis for the Nullspace or Column space of a matrix A.

Finding a basis for Col(A) when A is not in REF form..

Coordinate Systems From Non-Standard Bases | Definitions + Visualization.

Writing Vectors in a New Coordinate System **Example**.

What Exactly are Grid Lines in Coordinate Systems?.

The Dimension of a Subspace | Definition + First Examples.

Computing Dimension of Null Space & Column Space.

The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!.

Changing Between Two Bases | Derivation + Example.

Visualizing Change Of Basis Dynamically.

Example: Writing a vector in a new basis.

What eigenvalues and eigenvectors mean geometrically.

Using determinants to compute eigenvalues & eigenvectors.

Example: Computing Eigenvalues and Eigenvectors.

A range of possibilities for eigenvalues and eigenvectors.

Diagonal Matrices are Freaking Awesome.

How the Diagonalization Process Works.

Compute large powers of a matrix via diagonalization.

Full Example: Diagonalizing a Matrix.

COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**.

Visualizing Diagonalization & Eigenbases.

Similar matrices have similar properties.

The Similarity Relationship Represents a Change of Basis.

Dot Products and Length.

Distance, Angles, Orthogonality and Pythagoras for vectors.

Orthogonal bases are easy to work with!.

Orthogonal Decomposition Theorem Part 1: Defining the Orthogonal Complement.

The geometric view on orthogonal projections.

Orthogonal Decomposition Theorem Part II.

Proving that orthogonal projections are a form of minimization.

Using Gram-Schmidt to orthogonalize a basis.

Full example: using Gram-Schmidt.

Least Squares Approximations.

Reducing the Least Squares Approximation to solving a system.

### Taught by

Dr. Trefor Bazett