Linear Algebra
part 3 of 3

A total of 50 hours of lectures

This is an academic level course for university and college engineering. Due to its size, it is divided into three parts. This page describes the third of those three parts.

Level - Intermediate

  • High-school and college mathematics (mainly arithmetics, some trigonometry, polynomials)
  • Linear Algebra and Geometry 1 (systems of equations, matrices and determinants, vectors and their products, analytic geometry of lines and planes)
  • Linear Algebra and Geometry 2 (vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors, diagonalization)
  • Some basic calculus
    Basic knowledge of complex numbers (this course contains a short introduction to complex numbers)

Curriculum

Also make sure that you check with your professor what parts of the course you will need for your exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

Linear Algebra, part 3 of 3

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Get the outline

A detailed list of all the lectures in part 3 of the course, including which theorems will be discussed and which problems will be solved. If you are looking for a particular kind of problem or a particular concept, this is where you should look first.

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Course Objectives & Outcomes for part 3

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How to solve problems in linear algebra and geometry (illustrated with 144 solved problems) and why these methods work.
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Use diagonalization of matrices for solving various problems from different branches of mathematics (ODE, dynamical systems).
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Work with geometric concepts as length (norm), distance, angles, and orthogonality in non-geometric setups.
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Orthogonal and orthonormal bases, and Gram-Schmidt process in various inner product spaces.
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Symmetric matrices and their properties; orthogonal diagonalization: how it is done and how to understand it geometrically.
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Quadratic forms and their connection to symmetric matrices: uniqueness of this correspondence and its consequences.
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Some concepts from abstract algebra: group, ring, field, and isomorphism; understand the concept of isomorphic vector spaces.
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Note: all the vector spaces discussed in this course are spaces over R (not over the field of complex numbers), and all our matrices have only real entries.
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Solve more advanced problems on eigendecomposition and orthogonality than in the second course.
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Inner product spaces different from R^n: space of continuous functions, spaces of polynomials, spaces of matrices.
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Pythagorean Theorem, Cauchy-Schwarz inequality, and triangle inequality in various inner product spaces.
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Min-max problems using Cauchy-Schwarz inequality, Best Approximation Theorem, least squares solutions.
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Positive/negative definite matrices, indefinite matrices; various methods of determining definiteness of matrices.
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Geometry of quadratic forms in two and three variables: conic sections and quadratic surfaces.
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Crowning of the course and a natural consequence of all the other topics: Singular Value Decomposition and pseudoinverses.