About Course
In this course, We will explain the essentials of Linear Algebra (LA) and everything that you need to understand the basics of linear algebra. We will cover content such as:
 Matrices and Linear System of Equations
 Gaussian Elimination
 Reduced Echelon Form and RREF
 Matrix Algebra
 Special Matrices, Diagonal Matrices, and Inverse Matrices
 Inverse Matrices and The Inverses of Transposed Matrices
 Determinants and computing the Determinant
 Much more!
By the end of this course, you should very comfortable with linear algebra and be able to follow throw any Math which uses the (LA) notation.
You will also get answers to any questions on any (LA) questions that you might have for life.
Course Content
Linear Algebra

Determinants can be computed along any row or column – choose the easiest!
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What Exactly are Grid Lines in Coordinate Systems?
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The Dimension of a Subspace  Definition + First Examples
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Computing Dimension of Null Space & Column Space
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The Dimension Theorem  Dim(Null(A)) + Dim(Col(A)) = n  Also, Rank!
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Changing Between Two Bases  Derivation + Example
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Visualizing Change Of Basis Dynamically
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Example: Writing a vector in a new basis
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What eigenvalues and eigenvectors mean geometrically
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Writing Vectors in a New Coordinate System **Example**
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Coordinate Systems From NonStandard Bases  Definitions + Visualization
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Vector Spaces  Definition & Examples
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The Vector Space of Polynomials: Span, Linear Independence, and Basis
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Subspaces are the Natural Subsets of Linear Algebra  Definition + First Examples
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The Span is a Subspace  Proof + Visualization
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The Null Space & Column Space of a Matrix  Algebraically & Geometrically
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The Basis of a Subspace
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Finding a Basis for the Nullspace or Column space of a matrix A
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Finding a basis for Col(A) when A is not in REF form.
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Using determinants to compute eigenvalues & eigenvectors
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Example: Computing Eigenvalues and Eigenvectors
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A range of possibilities for eigenvalues and eigenvectors
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Orthogonal bases are easy to work with!
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Orthogonal Decomposition Theorem Part 1: Defining the Orthogonal Compliment
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The geometric view on orthogonal projections
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Orthogonal Decomposition Theorem Part II
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Proving that orthogonal projections are a form of minimization
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Using GramSchmidt to orthogonalize a basis
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Full example: using GramSchmidt
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Least Squares Approximations
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Distance, Angles, Orthogonality and Pythagoras for vectors
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Dot Products and Length
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Diagonal Matrices are Freaking Awesome
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How the Diagonalization Process Works
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Compute large powers of a matrix via diagonalization
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Full Example: Diagonalizing a Matrix
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COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**
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Visualizing Diagonalization & Eigenbases
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Similar matrices have similar properties
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The Similarity Relationship Represents a Change of Basis
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Reducing the Least Squares Approximation to solving a system
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What’s the big idea of Linear Algebra?
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How to determine if one vector is in the span of other vectors?
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MatrixVector Multiplication and the equation Ax=b
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MatrixVector Multiplication Example
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Proving Algebraic Rules in Linear Algebra — Ex: A(b+c) = Ab +Ac
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The Big Theorem, Part I
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Writing solutions to Ax=b in vector form
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Geometric View on Solutions to Ax=b and Ax=0.
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Three nice properties of homogeneous systems of linear equations
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Introducing Linear Combinations & Span
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What is a vector?
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What is a Solution to a Linear System?
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Visualizing Solutions to Linear Systems
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Rewriting a Linear System using Matrix Notation
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Using Elementary Row Operations to Solve Systems of Linear Equations
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Using Elementary Row Operations to simplify a linear system
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Examples with 0, 1, and infinitely many solutions to linear systems
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Row Echelon Form and Reduced Row Echelon Form
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Back Substitution with infinitely many solutions
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Linear Dependence and Independence – Geometrically
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Determining Linear Independence vs Linear Dependence
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Making a Math Concept Map  Ex: Linear Independence
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Visualizing Composition of Linear Transformations
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Elementary Matrices
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You can “invert” matrices to solve equations…sometimes!
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Finding inverses to 2×2 matrices is easy!
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Find the Inverse of a Matrix
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When does a matrix fail to be invertible? Also more “Big Theorem”.
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Visualizing Invertible Transformations (plus why we need onetoone)
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Invertible Matrices correspond with Invertible Transformations **proof**
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Computing matrix multiplication
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The motivation and definition of Matrix Multiplication
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Transformations and Matrix Transformations
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Three examples of Matrix Transformations
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Linear Transformations
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Are Matrix Transformations and Linear Transformation the same? Part I
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Every vector is a linear combination of the same n simple vectors!
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Matrix Transformations are the same thing as Linear Transformations
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Finding the Matrix of a Linear Transformation
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Onetoone, Onto, and the Big Theorem Part II
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Determinants – a “quick” computation to tell if a matrix is invertible
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