Linear algebra is central to almost all areas of mathematics. Linear Algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models. For nonlinear systems, which cannot be modeled with linear algebra, linear algebra is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.

Linear algebra is central to almost all areas of mathematics. For instance,

Linear Algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models. For nonlinear systems, which cannot be modeled with linear algebra, linear algebra is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.

*** Topics of the course ***

- Linear Algebra - Least Squares Approximation - 01 - Introduction (0:00)
- Linear Algebra - Least Squares Approximation - 02 - Fundamental Theorem (09:12)
- Linear Algebra - Least Squares Approximation - 03 - Fitting data to a straight curve Part 1 (21:17)
- Linear Algebra - Least Squares Approximation - 04 - Fitting data to a straight curve Part 2 (29:27)
- Linear Algebra - Least Squares Approximation - 05 - Fitting data to a straight curve Part 3 (37:40)
- Linear Algebra - Least Squares Approximation - 06 - Fitting data to a straight curve example (45:01)
- Linear Algebra - Least Squares Approximation - 07 - Fitting data to more general functions (51:10)
- Linear Algebra - Least Squares Approximation - 08 - The inverse of A transpose times A (01:01:38)
- Linear Algebra - Hamming's error correcting codes - 01 - Hamming matrices (1:13:31)
- Linear Algebra - Hamming's error correcting codes - 02 - Properties of Hamming matrices (01:17:35)
- Linear Algebra - Hamming's error correcting codes - 03 - Example (1:26:33)
- Linear Algebra - Hamming's error correcting codes - 04 - Parity bits (1:40:36)
- Topics in Linear Algebra - The Functional Calculus - 01 - Theorem and Example (1:50:47)
- Topics in Linear Algebra - The Functional Calculus - 02 - Square-root of a positive matrix (1:57:32)
- Topics in Linear Algebra - The Functional Calculus - 03 - Polynomial interpolation (2:04:18)
- Topics in Linear Algebra - The Functional Calculus - 04 - The determinant of a Van dermonde matrix (2:13:29)
- Topics in Linear Algebra - The Functional Calculus - 05 - Proof of main theorem (2:28:17)
- Affine subspaces and transformations - 01 - affine combinations (2:39:22)
- Affine subspaces and transformations - 02 - affine subspaces (2:49:08)
- Affine subspaces and transformations - 03 - affine transformations (3:13:11)
- Affine subspaces and transformations - 04 - composition of affine transformations (3:16:37)
- Stochastic maps - 01 - Conditional probabilities (3:25:53)
- Stochastic maps - 02 - Composing conditional probabilities (3:36:50)
- Stochastic maps - 03 - Products of conditional probabilities and a.e. equivalence (3:47:38)
- Stochastic maps - 04 -Bayes' theorem (3:58:37)

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