| 1. Lecture: Introduction to linear systems
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We look at the problem of solving systems of linear equations
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| 2. Lecture: matrices and Gauss-Jordan elimination
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We learn how to do Gauss-Jordan elimination for bringing a matrix into row reduced echelon form.
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| 3. Lecture: on solutions of linear systems
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There is a trichotomy: a system of linear equations has either 0,1 or infinitely many solutions.
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| 4. Lecture: Linear transformations and their inverses
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We define linear transformations and inversese
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| 5. Lecture: Linear transformations in geometry
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We deal with geometric representations of linear transformations
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| 6. Lecture: Matrix product and inverse
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Matrix algebra is like the algebra of numbers, but with a twist.
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| 7. Lecture: Image and kernel
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The image and kernel satisfy the rank-nullity theorem.
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| 8. Lecture: Basis and linear independence
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A basis is like the triple i,j,k frame in Euclidean space. It is linearly independent and spanning.
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| 9. Lecture: Dimension
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Abstract linear spaces and dimension are introduced.
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| 10. Lecture: Coordinates
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A basis defines coordinates. How do these coordinates change if the basis changes?
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| 11. Lecture: Linear spaces
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Linear spaces can also be function spaces
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| 12. Lecture: orthonormal bases and orthogonal projections
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Projections will be useful in statistics
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| 14. Lecture: Gram-Schmidt and QR factorization
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Gram-Schmidt allows to get an orthogonal basis from a given basis
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| 15. Lecture: Orthogonal transformations
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Orthogonal transformations can be rotations or reflections
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| 16. Lecture: Least squares and data fitting
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One of the most important applications of projections
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| 17. Lecture: Determinants I
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An important number attached to matrices.
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| 18. Lecture: Determinants II
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How does one compute determinants fast?
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| 19. Lecture: Eigenvalues
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Eigenvalues are numbers attached to a matrix which are coordinate independent.
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| 20. Lecture: Eigenvectors
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Eigenvector computations is one reason why we looked at the kernel
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| 21. Lecture: Diagonalization
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Many matrices are diagonal when looked at in the right basis.
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| 22. Lecture: Complex eigenvalues
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A bit of complex number
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| 24. Lecture: Stability
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Eigenvalues are related to stability
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| 25. Lecture: Symmetric matrices
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Symmetric matrices can be diagonalized
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| 26. Lecture: Differential equations I
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Ordinary differential equations can be solved by diagonalization
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| 27. Lecture: Differential equations II
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Or by using operators
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| 28. Lecture: Nonlinear systems
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Nonlinear systems are important in applications
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| 29. Lecture: operators on function spaces (11/17/2010)
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Linear operators
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| 30. Lecture: inhomogeneous differential equations
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Can be used to solve systems.
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| 31. Lecture: inhomogeneous differential equations (11/22/2010)
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Or systems p(D) f = g.
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| 32. Lecture: inner product spaces
A preparation for Fourier
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| 33. Lecture: Fourier series
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A highlight
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| 34. Lecture: Parseval's identity
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Its like pythagoras
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| 35. Lecture: Partial differential equations
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We learn how to solve PDEs
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