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What’s the big idea of Linear Algebra?00:00:00

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What is a Solution to a Linear System?00:00:00

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Visualizing Solutions to Linear Systems00:00:00

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Rewriting a Linear System using Matrix Notation00:00:00

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Using Elementary Row Operations to Solve Systems of Linear Equations00:00:00

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Using Elementary Row Operations to simplify a linear system00:00:00

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Examples with 0, 1, and infinitely many solutions to linear systems00:00:00

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Row Echelon Form and Reduced Row Echelon Form00:00:00

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Back Substitution with infinitely many solutions00:00:00

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What is a vector?00:00:00

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Introducing Linear Combinations & Span00:00:00

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How to determine if one vector is in the span of other vectors?00:00:00

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Matrix-Vector Multiplication and the equation Ax=b00:00:00

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Matrix-Vector Multiplication Example00:00:00

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Proving Algebraic Rules in Linear Algebra — Ex: A(b+c) = Ab +Ac00:00:00

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The Big Theorem, Part I00:00:00

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Writing solutions to Ax=b in vector form00:00:00

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Geometric View on Solutions to Ax=b and Ax=0.00:00:00

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Three nice properties of homogeneous systems of linear equations00:00:00

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Linear Dependence and Independence – Geometrically00:00:00

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Determining Linear Independence vs Linear Dependence00:00:00

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Making a Math Concept Map | Ex: Linear Independence00:00:00

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Transformations and Matrix Transformations00:00:00

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Three examples of Matrix Transformations00:00:00

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Linear Transformations00:00:00

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Are Matrix Transformations and Linear Transformation the same? Part I00:00:00

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Every vector is a linear combination of the same n simple vectors!00:00:00

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Matrix Transformations are the same thing as Linear Transformations00:00:00

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Finding the Matrix of a Linear Transformation00:00:00

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One-to-one, Onto, and the Big Theorem Part II00:00:00

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The motivation and definition of Matrix Multiplication00:00:00

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Computing matrix multiplication00:00:00

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Visualizing Composition of Linear Transformations00:00:00

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Elementary Matrices00:00:00

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You can “invert” matrices to solve equations…sometimes!00:00:00

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Finding inverses to 2×2 matrices is easy!00:00:00

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Find the Inverse of a Matrix00:00:00

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When does a matrix fail to be invertible? Also more “Big Theorem”.00:00:00

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Visualizing Invertible Transformations (plus why we need one-to-one)00:00:00

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Invertible Matrices correspond with Invertible Transformations **proof**00:00:00

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Determinants – a “quick” computation to tell if a matrix is invertible00:00:00

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Determinants can be computed along any row or column – choose the easiest!00:00:00

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Vector Spaces | Definition & Examples00:00:00

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The Vector Space of Polynomials: Span, Linear Independence, and Basis00:00:00

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Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples00:00:00

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The Span is a Subspace | Proof + Visualization00:00:00

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The Null Space & Column Space of a Matrix | Algebraically & Geometrically00:00:00

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The Basis of a Subspace00:00:00

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Finding a Basis for the Nullspace or Column space of a matrix A00:00:00

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Finding a basis for Col(A) when A is not in REF form.00:00:00

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Coordinate Systems From Non-Standard Bases | Definitions + Visualization00:00:00

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Writing Vectors in a New Coordinate System **Example**00:00:00

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What Exactly are Grid Lines in Coordinate Systems?00:00:00

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The Dimension of a Subspace | Definition + First Examples00:00:00

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Computing Dimension of Null Space & Column Space00:00:00

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The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!00:00:00

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Changing Between Two Bases | Derivation + Example00:00:00

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Visualizing Change Of Basis Dynamically00:00:00

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Example: Writing a vector in a new basis00:00:00

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What eigenvalues and eigenvectors mean geometrically00:00:00

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Using determinants to compute eigenvalues & eigenvectors00:00:00

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Example: Computing Eigenvalues and Eigenvectors00:00:00

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A range of possibilities for eigenvalues and eigenvectors00:00:00

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Diagonal Matrices are Freaking Awesome00:00:00

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How the Diagonalization Process Works00:00:00

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Compute large powers of a matrix via diagonalization00:00:00

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Full Example: Diagonalizing a Matrix00:00:00

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COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**00:00:00

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Visualizing Diagonalization & Eigenbases00:00:00

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Similar matrices have similar properties00:00:00

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The Similarity Relationship Represents a Change of Basis00:00:00

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Dot Products and Length00:00:00

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Distance, Angles, Orthogonality and Pythagoras for vectors00:00:00

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Orthogonal bases are easy to work with!00:00:00

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Orthogonal Decomposition Theorem Part 1: Defining the Orthogonal Compliment00:00:00

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The geometric view on orthogonal projections00:00:00

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Orthogonal Decomposition Theorem Part II00:00:00

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Proving that orthogonal projections are a form of minimization00:00:00

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Using Gram-Schmidt to orthogonalize a basis00:00:00

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Full example: using Gram-Schmidt00:00:00

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Least Squares Approximations00:00:00

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Reducing the Least Squares Approximation to solving a system00:00:00