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.