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su f mastering linear algebra introduction applications 2019

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Textbook in PDF format Linear algebra may well be the most accessible of all routes into higher mathematics. It requires little more than a foundation in algebra and geometry, yet it supplies powerful tools for solving problems in subjects as diverse as computer science and chemistry, business and biology, engineering and economics, and physics and statistics, to name just a few. Furthermore, linear algebra is the gateway to almost any advanced mathematics course. Calculus, abstract algebra, real analysis, topology, number theory, and many other fields make extensive use of the central concepts of linear algebra: vector spaces and linear transformations. Professor Biography Course Scope Transformations Vectors Linear Combinations Abstract Vector Spaces The Dot Product Properties of the Dot Product A Geometric Formula for the Dot Product The Cross Product Describing Lines Describing Planes What Is a Matrix? Matrix Multiplication The Identity Matrix Other Matrix Properties Multivariable Functions Definition of a Linear Transformation Properties of Linear Transformations Matrix Multiplication Is a Linear Transformation Examples of Linear Transformations Linear Equations Systems of Linear Equations Solving Systems of Linear Equations Gaussian Elimination Getting Infinitely Many or No Solutions Quiz for Lectures 1–6 Reduced Row Echelon Form Using the RREF to Find the Set of Solutions Row-Equivalent-Matrices 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 The Null-Space of a Matrix Subspaces The Row Space and Column Space of a Matrix 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 The Inverse of a Matrix Finding the Inverse of a 2 × 2 Matrix Properties of Inverses 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 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 Population Dynamics Application Understanding Matrix Powers Eigenvectors and Eigenvalues Solving the Eigenvector Equation Return to Population Dynamics Application Eigenvectors and Eigenvalues: Geometry The Geometry of Eigenvectors and Eigenvalues Verifying That a Vector Is an Eigenvector Finding Eigenvectors and Eigenvalues Matrix Powers Change of Basis Eigenvalues and the Determinant Algebraic Multiplicity and Geometric Multiplicity Diagonalizability Similar Matrices Recalling the Population Dynamics Model High Predation Low Predation Medium Predation Differential Equations: New Applications Solving a System of Differential Equations Complex Eigenvalues Quiz for Lectures 13–18 Orthogonal Sets Orthogonal Matrices Properties of Orthogonal Matrices The Gram-Schmidt Process QR-Factorization Orthogonal Diagonalization Markov Chains: Hopping Around Markov Chains Economic Mobility Theorems about Markov Chains Single-Variable Calculus Multivariable Functions Differentiability The Derivative Chain Rule Multilinear Regression: Least Squares Linear Regression Multiple Linear Regression Invertibility of the Gram Matrix How Good Is the Fit? Polynomial Regression The Singular Value Decomposition The Geometric Meaning of the SVD Computing the SVD 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

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