Introduction to Linear Algebra

As Taught in:

2018/2019

Level:

Undergraduate

Learning Resource Types:

=> Problem Sets

=> Notes from instructor insights

=>Reading Resources

Course Description:

Introduction to Linear Algebra is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.

Course Meeting Times:

Lectures: 3 sessions of 1 hour / week

Prerequisites:

Multivariable Calculus (Calculus 102)

Text:

The readings are assigned inside tab “Readings”

Goals:

The goals for this course – Introduction to Linear Algebra – are using matrices and also understanding them.

Here are some key computations and some of the ideas behind them:

1. Solving “Ax=b” for square systems by elimination (pivots, multipliers, back substitution, invertibility of A, factorization into A=LU)
2. Complete solution to “Ax=b” – column space containing “b”, rank of “A”, nullspace of “A”, and special solutions to “Ax=0 from row reduced “R”)
3. Basis and dimension – bases for the four fundamental subspaces
4. Least squares solutions – closest line by understanding projections
5. Orthogonalization by Gram-Schmidt – factorization into “A=QR”
6. Properties of determinants – leading to the cofactor formula and the sum over all “n!” permutations, applications to “inv(A)” and Volume.
7. Eigenvalues and Eigenvectors – diagonalizing “A”, computing powers “A^k” and matrix exponentials to solve difference and differential equations.
8. Symmetric matrices and positive definite matrices – real eigenvalues and orthogonal eigenvectors, tests for “x’Ax>0”, applications.
9. Linear transformations and change of basis – connected to the Singular Value Decomposition – orthonormal bases that diagonalize “A”.
10. Linear algebra in Engineering – graphs and networks, Markov Matrices, Fourier Matrix, Fast Fourier Transform, Linear Programming.

Practices

The practices are essential in learning linear algebra. They are not a test, and you are encouraged to talk to other learners about difficult problems. Especially after you have found them challenging. Talking about Linear Algebra is healthy. However, you must write your own solutions.

Study Materials

The textbook for this course from:

Strang, Gilbert. 2009. Introduction to Linear Algebra 4th Ed. Published by Wellesly-Cambridge Press. ISBN: 9780980232714

MATLAB

The use of calculators or notes is not permitted, however some assignment problems will require you to use MATLAB, an important tool for mastering numerical linear algebra.

No previous MATLAB experience is required in this course.

The “related resources” tab has links to information about MATLAB, including a tutorial.

A topic that will be covered are:

1. The geometry of linear equations
2. Elimination with matrices
3. Matrix operations and inverses
4. LU and LDU factorizations
5. Transposes and permutations
6. Vector spaces and subspaces
7. The nullspace: Solving “Ax=0”
8. Rectangular “PA=LU” and “Ax=b”
9. Row reduced echelon form
10. Basis and dimension
11. The four fundamental subspaces
12. Graphs and networks
13. Orthogonality
14. Projections and subspaces
15. Least squares approximations
16. Gram-Schmidt and “A=QR”
17. Properties of determinants
18. Formulas of determinants
19. Applications of determinants
20. Eigenvalues and Eigenvectors
21. Diagonalization
22. Markov Matrices
23. Differential Equations
24. Symmetric Matrices
25. Positive Definite Matrices
26. Matrices in Engineering
27. Similar Matrices
28. Singular Value Decomposition
29. Fourier Series, FFT, Complex Matrices
30. Linear Transformations
31. Choice of Basis
32. Linear Programming
33. Numerical Linear Algebra
34. Computational Science

Before we start, please have an attention to the “NOTE” below.

NOTE: More material on linear algebra (and much more about differential equations) is in 2014 textbook “Differential Equations and Linear Algebra“. Also, in the version of 2016 textbook of “Learn Differential Equations.”

Textbook:

Strang, Gilbert. 2009. Introduction to Linear Algebra 4th Ed. Wellesley-Cambridge Press. ISBN: 9780980232714.

Section 1.1 – 2.1 : The Geometry of Linear Equations

Section 2.2 – 2.3 : Elimination with Matrices

Section 2.4 – 2.5 : Matrix Operations and Inverses

Section 2.6 : LU and LDU factorization

Section 2.7 : Transposes and Permutations

Section 3.1 : Vector spaces and subspaces

Section 3.2 : The nullspace: Solving “Ax=0”

Section 3.3 – 3.4 : Rectangular “PA=LU” and “Ax=b” and Row reduced echelon form.

Section 3.5 : Basis and dimension

Section 3.6 : The Four fundamental subspaces

Section 8.2 : Graphs and networks

Section 4.1 : Orthogonality

Section 4.2 : Projections and subspaces

Section 4.3 : Least squares approximations

Section 4.4 : Gram-Schmidt and “A=QR”

Section 5.1 : Properties of determinants

Section 5.2 : Formulas for determinants

Section 5.3 : Applications for determinants

Section 6.1 : Eigenvalues and Eigenvectors

Section 6.2 : Diagonalization

Section 8.3 : Markov Matrices

Section 6.3 : Differential Equations

Section 6.4 : Symmetric Matrices

Section 6.5 : Positive definite matrices

Section 8.1 : Matrices in Engineering

Section 6.6 : Similar Matrices

Section 6.7 : Singular Value Decomposition

Section 8.5, 10.2 – 10.3 : Fourier Series, FFT, Complex Matrices

Section 7.1 – 7.2 : Linear Transformations

Section 7.3 : Choice of Basis

Section 8.4 : Linear Programming

Section 9.1 – 9.3 : Numerical Linear Algebra

The practices will help you to understand more in this course “Introduction to Linear Algebra“. Please do login with your education username and password to start with your practices.

This problem solving section will help you with much better understanding of your learning in “Introduction to Linear Algebra“. Please login with your education username and password to get start with it.