Linear Algebra

You are here

Course Information
Terms offered: 
Language of instruction: 
Contact Hours: 

Two semesters of College Calculus/Instructor’s consent.


Linear Algebra is the branch of mathematics that involves the study of solving systems of Linear Equations, vector spaces and linear mappings. The mastery of the subject is essential to any major related to applied and pure Mathematics and in particular to Engineering.

In this course, students will become proficient in the techniques of solving and manipulating systems. A particular focus of the course will be upon the applications of linear algebra to various Engineering fields. Among the most salient goals of the course will be an in-depth understanding of the primary theories as to why the techniques function as they do.

In particular, the course content covered during the semester is: Matrices, Determinants, Gaussian elimination, vector spaces, Linear Transformations, LU-decomposition, orthogonality, Gram-Schmidt process, inner products, eigenvalue problems, applications to differential equations and Markov processes.

Attendance policy: 

Attendance is mandatory for all IES Abroad classes, including course-related excursions. Any exams, tests, presentations, or other work missed due to student absences can only be rescheduled in cases of documented medical or family emergencies. If a student misses more than two classes in any course half a letter grade will be deducted from the final grade for every additional absence. Seven absences in any course will result in a failing grade.

Learning outcomes: 

On successful completion of the course, students should be able to:

  • Solve systems of linear equations, manipulate matrix algebra and determinants, apply row operations and elementary matrices;
  • Analyze the concept and basic structure of vector spaces, give examples of vector spaces and examples of sets endowed with operations that are not vector spaces, explain the concept of dimension, apply the dimension theorem.
  • Elucidate the nullspace, row space and column space of a matrix, apply the rank-nullity theorem.
  • Give examples of linear transformations, determine whether or not a map is linear, evaluate the matrix representations of a linear transformation.
  • Evaluate eigenvalues and eigenvectors, evaluate algebraic multiplicity and geometric multiplicity, diagonalize a matrix.
  • Write simple proofs of statements related to the theory covered in the course.
Method of presentation: 

Three hours of lecture per week (the instructor will present the theory and cover examples) and one hour of recitation per week to do exercises, examples and prepare homework. There will be two class visits related to the applications of the content of the course to Engineering: a visit to the NASA station in Robledo de Chavela, and a visit to Animation Studios in Madrid.

Required work and form of assessment: 

The final grade for the course will consist of:

  • Homework (10%): Homework will be assigned each week. No late homework will be accepted. Homework is due at the beginning of the lecture. The two lowest homework scores will be deleted for grading purposes.
  • Class presentation (10%): Presentation an application of Linear Algebra to Engineering assigned by the instructor on the third week of class.
  • Midterm exam (40%)
  • Final exam (40%): The final exam is comprehensive and will include a bonus question worth 10/100 extra points that will allow the students to make up a low midterm score.

(Section 1.1)



Lecture 1 Linear Systems of Equations and Row Operations
Lecture 2
(Section 1.2)
Row-Echelon Form. Gauss-Jordan operations
Session 1

Section 1.1, exercises 2b, 2ad, 4a, 5b, 6b, 9, 10, 11

Section 1.2, exercises 5ad, 10, 13, 15

Lecture 3
(Section 1.3)

Matrix Arithmetic

​Homework 1 is due at the beginning of the lecture

Lecture 4
(Section 1.4)
Matrix Algebra
Session 2

Section 1.3, exercises 6, 7, 13, 15, 16, 17

Section 1.4, exercises 3, 4, 5, 7, 10a, 18, 22, 24a, 31, 32, 33

Lecture 5
(Section 1.5)
Elementary matrices; LU-decomposition ​Homework 2 is due at the beginning of the lecture
Lecture 6
(Sections 2.1-2.2)
Determinants. Computation by elimination and/or cofactors. Properties of determinants.
Session 3

Section 1.5, exercises 3a, 6, 9, 10ae, 28

Section 2.1, exercises 4ac, 5, 5, 11 

Section 2.2, exercises 4a, 7, 12

Lecture 7
(Section 2.3)

Cramer's Rule. Additional topics and applications: coding messages.

Homework 3 is due at the beginning of the lecture.

​Lecture 8
(Section 3.1)
Vector Spaces, definition and examples.
Session 4

Section 2.3, exercises 1d, 2ab, 5, 14

Section 3.1, exercises 1, 4, 5, 6

Lecture 9
(Section 3.2)
Vector Subspaces. Span of a set of vectors ​Homework 4 is due at the beginning of the lecture
​Lecture 10
(Section 3.3)
Linear independence
Session 5

Section 3.2, exercises 1ad, 2ac, 3, 4ab, 11ab, 16a, 20

Section 3.3, exercises 1ac, 2ab, 3ab, 5, 9

Lecture 11
(Section 3.4)

Basis and dimension

Homework 5 is due at the beginning of the lecture

Lecture 12
(Section 3.5)
Change of Basis. Change of coordinates.
Session 6

Section 3.4, exercises 3, 4, 5, 9, 11

Section 3.5, exercises 1a, 2a, 4, 6, 9

Lecture 13
(Section 3.6)

Row space and column space. Review for Midterm.

Homework 6 is due at the beginning of the lecture.

Session 7

Section 3.6, exercises 1a, 2a, 4abc, 5abc

Lecture 14 Midterm 1

Lecture 15
(Section 4.1)

Linear Transformations, definitions and examples. Image and Kernel.  Homework 7 is due at the beginning of the lecture.
Lecture 16
(Section 4.2)
Matrix representation of a Linear Transformation. Application: Computer Graphics and animation.
Session 8

Section 4.1, exercises 2, 3, 5, 7ab, 19, 20.

Section 4.2, exercises 2a, 3b, 5, 10a, 11a, 14b

Lecture 17
(Section 4.3)
Similarity. Application: Computer Graphics and animation. Homework 8 is due at the beginning of the lecture.
Lecture 18
(Section 5.1)
The scalar product in real vector spaces. Application: Information retrieval.
Session 9

Section 4.3, exercises 1a, 2, 5, 7, 14

Section 5.1, exercises 1ab, 2ab, 5, 10, 17

Lecture 19
(Section 5.2)

Orthogonal subspaces.

​Homework 9 is due at the beginning of the lecture.

​Lecture 20
(Section 5.3)
Least squares problems. Application to Astronomy.
Session 10

Section 5.2, exercises 1ac. 2, 3, 4, 12

Section 5.3, exercises 1a, 2a, 3a, 5, 14

Lecture 21
(Section 5.4)

Inner product spaces. Cauchy-Schwarz inequality.

Homework 10 is due at the beginning of the lecture.

Lecture 22
(Section 5.5)
Orthonormal basis and projection. Application: Signal Processing. The Discrete Fourier Transform.
Session 11

Section 5.4, exercises 1, 2, 3, 4, 10

Section 5.5, exercises 2, 3, 4, 6, 8, 21

Lecture 23
(Sections 5.6 & 5.7)

The Gram-Schmidt orthogonalization process and QR-factorization. Orthogonal Polynomials.

Homework 11 is due at the beginning of the lecture.

Lecture 24
(Section 6.1)
Eigenvalues and eigenvectors. Application to Aerospace: The orientation of a space shuttle. Similar matrices (review).
Session 12

Section 5.6, exercises 1, 2, 5, 8

Section 5.7, exercises 2, 6, 9

Section 6.1, exercises 1af, 2, 3, 4, 10

Lecture 25
(Section 6.2)

Systems of linear differential equations. Application: Vibrations of a building.​

Homework 12 is due at the beginning of the lecture

Lecture 26
(Section 6.3)
Diagonalization. Applications: Markov Chains and web searches & page ranking.
Session 13

Section 6.2, exercises 1ai, 2a, 3, 5a, 6

Section 6.3, exercises 1ae, 2ae, 3ae, 4a, 8a, 22, 24

Lecture 27
(Section 6.4)

Hermitian matrices. Shur’s theorem, spectral theorem. Real symmetric matrices. Normal matrices.

Homework 13 is due at the beginning of the lecture.

​Lecture 28
(Section 6.5)
The singular value decomposition. Application: Digital Image processing and information retrieval.
Session 14

Section 6.4, exercises 1a, 2, 4ae, 5ab, 6, 17

Section 6.5, exercises 1, 2, 5

Lecture 29
(Section 6.6)
Quadratic forms. Review for final.
Session 15
Review for final exam


Required readings: 
  • ''Linear Algebra with Applications'' (8th edition-International Edition) by Steven J. Leon University of Massachusetts Dartmouth, Prentice-Hall, 2009. ISBN-13: 978-013512867-1