Linear Algebra and Analytical Geometry II

Academic year
Subject Area
Language of Instruction
Mode of Delivery
ECTS Credits
1st Cycle Studies

Recommended Prerequisites

Linear Algebra and Analytic Geometry I.

Teaching Methods

Theoretical-practical classes are mainly expository and include examples and exercises to apply the material being taught. These activities are mainly centered on the teacher.

In practical classes students are given exercises and examples which should be worked on during classes with the help of the teacher.

Extensive tutorial time is offered to students to solve their difficulties with the theory and to help them with  homework assignments and preparation for the exams.

Learning Outcomes

The main goal of this curricular unit is to teach  students to work with abstract vector spaces (finite and infinite dimensional) and linear maps. In this curricular unit, students are introduced to infinte dimensional vector spaces. The generalization to infinite dimensional vector spaces of results studied in ALGA I will play a fundamental role in the development of  the mathematical abstraction skills required to a student of an undergraduate course in Mathematics.

The main skills students are expected to develop are: knowledge of mathematical results; ability for generalization and abstraction; ability to formulate and solve mathematical problems; logical reasoning; correct mathematical writing; individual initiative; researching capacity; autonomous learning ability; independent thinking; communication skills.

Work Placement(s)



1. Eigenvalues and eigenvectors of matrices.  Matrix diagonalization.  Similarity of matrices.  Diagonalization of real symmetric matrices. Conics and quadrics.

2. Fields.

3. Vector spaces: subspaces; linear independence; bases and dimension; change of basis matrix.

4. Linear maps. Kernel and image. Isomorphisms.  Matrix of a linear map.  Eigenvalues and eigenvectors of a linear map.

5. Real inner product vector spaces.  Orthogonal projection.  Applications.

Head Lecturer(s)

Cristina Helena de Matos Caldeira

Assessment Methods

Approval in this course unit requires to score at least 10 (out of 20). The students that perform, along the semester, the mid-term exams and the tests may be exempted from final examination. The sum of percentages corresponding to these two components is 100%. The other students have to be evaluated at the end of the semester through a written examination (exam). : 100.0%


A. P. Santana,  J. F. Queiró, Introdução à Álgebra Linear, 2010