Linear Algebra

Dive into a comprehensive 10-hour introductory course on Linear Algebra, emphasizing both conceptual understanding and practical application of key techniques. Explore major topics including solving linear systems, linear transformations, linear independence, bases, dimension, eigenvalues and eigenvectors, diagonalization, orthogonality, and the Gram-Schmidt process. Begin with an introduction to the fundamental concepts and progress through visualizing solutions to linear systems, matrix operations, vector spaces, and subspaces. Master techniques for finding inverses, determinants, and bases for nullspaces and column spaces. Delve into coordinate systems, dimension theorems, and change of basis operations. Investigate eigenvalues and eigenvectors, both geometrically and algebraically, and learn the power of diagonalization. Conclude with an exploration of dot products, orthogonality, and least squares approximations, providing a solid foundation for further study in mathematics and related fields.

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.