Mastering Linear Algebra. An Introduction with Applications

Professor Biography Course Scope Lecture 1—Linear Algebra: Powerful Transformations Transformations Lecture 2—Vectors: Describing Space and Motion Vectors Linear Combinations Abstract Vector Spaces Lecture 3—Linear Geometry: Dots and Crosses The Dot Product Properties of the Dot Product A Geometric Formula for the Dot Product The Cross Product Describing Lines Describing Planes Lecture 4—Matrix Operations What Is a Matrix? Matrix Multiplication The Identity Matrix Other Matrix Properties Lecture 5—Linear Transformations Multivariable Functions Definition of a Linear Transformation Properties of Linear Transformations Matrix Multiplication Is a Linear Transformation Examples of Linear Transformations Lecture 6—Systems of Linear Equations Linear Equations Systems of Linear Equations Solving Systems of Linear Equations Gaussian Elimination Getting Infinitely Many or No Solutions Quiz for Lectures 1–6 Lecture 7—Reduced Row Echelon Form Reduced Row Echelon Form Using the RREF to Find the Set of Solutions Row-Equivalent-Matrices Lecture 8—Span and Linear Dependence The Span of a Set of Vectors When Is a Vector in the Span of a Set of Vectors? Linear Dependence of a Set of Vectors Linear Independence of a Set of Vectors Lecture 9—Subspaces: Special Subsets to Look For The Null-Space of a Matrix Subspaces The Row Space and Column Space of a Matrix Lecture 10—Bases: Basic Building Blocks Geometric Interpretation of Row, Column, and Null-Spaces The Basis of a Subspace How to Find a Basis for a Column Space How to Find a Basis for a Row Space How to Find a Basis for a Null-Space The Rank-Nullity Theorem Lecture 11—Invertible Matrices: Undoing What You Did The Inverse of a Matrix Finding the Inverse of a 2 × 2 Matrix Properties of Inverses Lecture 12—The Invertible Matrix Theorem The Importance of Invertible Matrices Finding the Inverse of an n × n Matrix Criteria for Telling If a Matrix Is Invertible Quiz for Lectures 7–12 Lecture 13—Determinants: Numbers That Say a Lot The 1 × 1 and 2 × 2 Determinants The 3 × 3 Determinant The n × n Determinant Calculating Determinants Quickly The Geometric Meaning of the n × n Determinant Consequences Lecture 14—Eigenstuff: Revealing Hidden Structure Population Dynamics Application Understanding Matrix Powers Eigenvectors and Eigenvalues Solving the Eigenvector Equation Return to Population Dynamics Application Lecture 15—Eigenvectors and Eigenvalues: Geometry The Geometry of Eigenvectors and Eigenvalues Verifying That a Vector Is an Eigenvector Finding Eigenvectors and Eigenvalues Matrix Powers Lecture 16—Diagonalizability Change of Basis Eigenvalues and the Determinant Algebraic Multiplicity and Geometric Multiplicity Diagonalizability Computing Matrix Powers Similar Matrices Lecture 17—Population Dynamics: Foxes and Rabbits Recalling the Population Dynamics Model High Predation Low Predation Medium Predation Lecture 18—Differential Equations: New Applications Solving a System of Differential Equations Complex Eigenvalues Quiz for Lectures 13–18 Lecture 19—Orthogonality: Squaring Things Up Orthogonal Sets Orthogonal Matrices Properties of Orthogonal Matrices The Gram-Schmidt Process QR-Factorization Orthogonal Diagonalization Lecture 20—Markov Chains: Hopping Around Markov Chains Economic Mobility Theorems about Markov Chains Lecture 21—Multivariable Calculus: Derivative Matrix Single-Variable Calculus Multivariable Functions Differentiability The Derivative Chain Rule Lecture 22—Multilinear Regression: Least Squares Linear Regression Multiple Linear Regression Invertibility of the Gram Matrix How Good Is the Fit? Polynomial Regression Lecture 23—Singular Value Decomposition: So Cool The Singular Value Decomposition The Geometric Meaning of the SVD Computing the SVD Lecture 24—General Vector Spaces: More to Explore Functions as Vectors General Vector Spaces Fibonacci-Type Sequences as a Vector Space Space of Functions as Vector Spaces Solutions of Differential Equations Ideas of Fourier Analysis Quiz for Lectures 19–24 Solutions Lectures 1–6 Lectures 7–12 Lectures 13–18 Lectures 19–24 Bibliography