| 1. | Introduction of the course, Applied Linear Algebra | Four Special Matrices | ||
| 2. | |
Applied Linear Algebra | Four Special Matrices Differences, Derivatives, Boundary condition |
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Applied Linear Algebra | Differences, Derivatives, Boundary condition | |
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| 3. | |
Applied Linear Algebra | Differences, Derivatives, Boundary condition Elimination leads to K = LDL^T |
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Applied Linear Algebra | Elimination leads to K = LDL^T Inverse and Delta function |
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| 4. | |
Applied Linear Algebra | Inverse and Delta function | |
| 5. | |
Applied Linear Algebra | Inverse and Delta function Eingenvalues and Eigenvectors |
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Applied Linear Algebra | Eingenvalues and Eigenvectors | |
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| 6. | |
Applied Linear Algebra (Eingenvalues and Eigenvectors) | Part 1 : Ax = \lambda x and A^k x = \lambda^k x and Diagonalizing A | |
| 7. | |
Applied Linear Algebra (Eingenvalues and Eigenvectors) | Part 2 : Eigenvectors for Derivatives and Differences | |
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Applied Linear Algebra (Eingenvalues and Eigenvectors, Positive Definite Matrices) | Part 2 : Eigenvectors for Derivatives and Differences, What is Positive Definite? | |
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| 8. | |
Applied Linear Algebra (Positive Definite Matrices) | Examples and Energy-based Definition, Positive definiteness, Minimum Problem in n Dimensions, Test for a Minmimum, Newtons Method | |
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Applied Linear Algebra (Positive Definite Matrices, Numerical Linear Algebra: LU, QR, SVD) | Three Essential Factorizations, Orthogonal Matrices, Orthogonalization A = QR, SVD | |
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| 9. | |
Applied Linear Algebra (Numerical Linear Algebra: LU, QR, SVD) | SVD, The pseudoinverse, Condition Numbers and Norms | |
| 10. | |
Applied Linear Algebra (Numerical Linear Algebra: LU, QR, SVD) | Condition Numbers and Norms, Mass Spring System, Stiffness Matrix and Solution | |
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A Framework for Applied Mathematics(Equilibrium and Stiffness Matrix) | Fixed End and Free End, Minimum Principles | |
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| 11. | |
A Framework for Applied Mathematics (Oscillation by Newtons Law) | One Mass and One Spring, Key Example: Motion Around a Circle (Four Finite Difference Methods) | |
| 12. | |
A Framework for Applied Mathematics (Oscillation by Newtons Law) | Key Example: Motion Around a Circle, Line of Masses and Springs | |
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A Framework for Applied Mathematics (Oscillation by Newtons Law) | Total Energy is Conserved, Applied Force and Resonance, Explicit Finite Differences, Stability and Instability | |
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| 13. | |
A Framework for Applied Mathematics (Oscillation by Newtons Law, Least Squares for Rectangular Matrices) | Explicit Finite Differences, Stability and Instability Linear Algebra Interpretation | |
| 14. | |
A Framework for Applied Mathematics (Least Squares for Rectangular Matrices, Graph Models and Kirchhoffs laws) | Linear Algebra Interpretation, Computation of Least Squares, The Incident Matrix | |
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A Framework for Applied Mathematics (Graph Models and Kirchhoffs laws) | The Graph Laplacian Matrix A^T*A, Inputs b, f and Matrices A, C, A^T, Assembling the Matrix K = A^T*C*A | |
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| 15. | |
A Framework for Applied Mathematics (Graph Models and Kirchhoffs laws, Networks and Transfer Functions) | KKT matrix, Impedance | |
| 16. | |
A Framework for Applied Mathematics (Networks and Transfer Functions) | Time Domain versus Frequency Domain, Transient Response and the Transfer Function, Underdamping and Overdamping | |
| 17. | |
A Framework for Applied Mathematics (Graph Models and Kirchhoffs laws, Networks and Transfer Functions) | KKT matrix, Impedance | |
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A Framework for Applied Mathematics (Networks and Transfer Functions) | Phase, Time Domain versus Frequency Domain, Transient Response and the Transfer Function | |
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| 18. | |
A Framework for Applied Mathematics (Networks and Transfer Functions) | Time Domain versus Frequency Domain, Transient Response and the Transfer Function, Underdamping and Overdamping | |
| 19. | |
Boundary Value Problems (Differential Equations and Finite Elements) | The Framework A^TCA for a Hanging Bar, General Solution and Examples, The Transpose A = d/dx | |
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Boundary Value Problems (Differential Equations and Finite Elements) | Galerkins Method, Comparison with Finite Difference, More accurate Finite Elements | |
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| 20. | |
Boundary Value Problems (Differential Equations and Finite Elements) | Galerkins Method, Comparison with Finite Difference, More accurate Finite Elements | |
| 21. | |
Boundary Value Problems (Cubic splines and Forth-Order Equations, Gradient and Divergence) | Cubic Finite Elements, Finite Difference for (cu)=f, Gradient, Divergence | |
| 22. | |
Boundary Value Problems (Gradient and Divergence) | Gradient, Divergence, Curl, Two important identities | |
| 23. | |
Boundary Value Problems (Gradient and Divergence) | Two important identies, Converse of the Two Identities, Harmonic Function | |
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Boundary Value Problems (Laplace Equation) | Complex Plane, Analytic function, Derivative of complex valued function, Cauchy-Riemann Equation | |
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| 24. | |
Boundary Value Problems (Laplace Equation) | Polar Coordinates: Laplace Equation in a Circle | |
| 25. | |
Boundary Value Problems (Laplace Equation, Finite Differences and Fast Poisson Solver) | Polar Coordinates: Laplace Equation in a Circle, Poissons Equation in a Square | |
| 26. | |
Boundary Value Problems (Finite Differences and Fast Poisson Solver, The Finite Element Method) | Elimination and Fill-in, Solver Using Eigenvalues, Fast Poisson Solvers, Trial and Test Functions: Galerkins method | |
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Boundary Value Problems (The Finite Element Method) | Pyramid Functions, Element Matricers and Element Vectors | |
| 28. | |
Boundary Value Problems(The Finite Element Method) | Global Matrix K and F frome K_e and F_e, Element Matrix in Two Dimension | |
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Boundary Value Problems(The Finite Element Method) | Element Matrix in Two Dimension, Quadrilateral Elements | |
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