Linear Algebra
part 2 of 3

A total of almost 47 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 second of those three parts.

Prerequisites

  • Linear Algebra and Geometry 1 (systems of equations, matrices and determinants, vectors and their products, analytic geometry of lines and planes)
  • High-school and college mathematics (mainly arithmetics, some trigonometry, polynomials)
  • Some basic calculus (used in some examples)
  • Basic knowledge of complex numbers (used in an example)

    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 2 of 3

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    Get the outline

    A detailed list of all the lectures in part 2 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.

    Get Linear Algebra part 2 on Udemy

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    Course Objectives & Outcomes for part 2

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    How to solve problems in linear algebra and geometry (illustrated with 153 solved problems) and why these methods work.
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    Linear combinations, linear dependence and independence in various vector spaces, and how to interpret them geometrically in R^2 and R^3.
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    Row space, columns space and nullspace for matrices, and about usage of these concepts for solving various types of problems.
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    How to compose linear transformations and how to compute their standard matrices in different bases; compute the kernel and the image for transformations.
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    Work with various geometrical transformations in R^2 and R^3, be able to compute their matrices and explain how these transformations work.
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    Transform any given basis for a subspace of R^n into an orthonormal basis of the same subspace with help of Gram-Schmidt Process.
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    Determine whether a given matrix is diagonalizable or not, and perform its diagonalization if it is.
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    Use diagonalization for problem solving involving computing the powers of square matrices, and motivate why this method works.
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    Use Wronskian to determine whether a set of smooth functions is linearly independent or not; be able to compute Vandermonde determinant.
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    Important concepts concerning vector spaces, such as basis, dimension, coordinates, and subspaces.
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    How to recalculate coordinates from one basis to another, both with help of transition matrices and by solving systems of equations.
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    Linear transformations: different ways of looking at them (as matrix transformations, as transformations preserving linear combinations).
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    Understand the connection between matrices and linear transformations, and see various concepts in accordance with this connection.
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    Understand the concept of isometry and be able to give some examples, and formulate their connection with orthogonal matrices.
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    Compute eigenvalues, eigenvectors, and eigenspaces for a given matrix, and give geometrical interpretations of these concepts.
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    Understand the relationship between diagonalizability and dimensions of eigenspaces for a matrix.
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    Be able to formulate and use The Invertible Matrix Theorem and recognise the situations which are suitable for the determinant test (and which are not).
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    Work with various vector spaces, for example with R^n, the space of all n-by-m matrices, the space of polynomials, the space of smooth functions.