# MATA122 Linear Algebra and Geometry 2 (4 cr)

**Study level:**

Intermediate studies

**Grading scale:**

0-5

**Language:**

English, Finnish

**Responsible organisation:**

Department of Mathematics and Statistics

**Curriculum periods:**

2020-2021, 2021-2022, 2022-2023, 2023-2024

## Description

Change of basis, eigenvalue theory, quadratic forms, abstract vector spaces and related concepts, sub-spaces, basis, dimension, linear transformations and matrices.

The contents correspond to

Lay: Linear algebra and its
applications (2nd ed.) chapters 4.1-4.7, 5.1-5.4, 6.5 and 7.1-7.2.

or

Anton ja Rorres: Elementary Linear Algebra (11th ed) chapters 4-8.

## Learning outcomes

After completing the course, the student

- knows the definitions of the eigenvalues, -vectors, and -spaces of a linear mapping and square matrix, and can find these for a given mapping and matrix
- is able to examine whether a given square matrix is diagonalizable
- is able to apply a quadratic form associated with a symmetric matrix to identify and recognize conic sections and quadratic surfaces
- Is familiar with the definitions of symmetric linear mapping and matrix, and the importance of symmetry in finding eigenvalues and -spaces and in diagonalizing
- is familiar with the notion of least-squares solution and knows how to compute it
- can give examples of real vector spaces and linear mappings between them
- knows the connection between linear mappings between finite dimensional vector spaces and matrices and can use this connection
- can calculate how changing bases changes the vector coordinates and the matrix corresponding to the linear mapping
- is familiar with the general definition of an inner product space and is able to investigate whether a given mapping is an inner product
- is able to determine the orthogonality and orthonormality of a vector set by using the inner product
- is able to use Geogebra to familiarize with geometric concepts in the course

## Description of prerequisites

Linear algebra and geometry 1

## Study materials

Lecture notes (in Finnish)

## Completion methods

### Method 1

**Evaluation criteria:**

Grading is based on points from the course exam and weekly exercises

**Time of teaching:**

Period 3

Select all marked parts

### Method 2

**Evaluation criteria:**

Grading is based on points from the final exam

Select all marked parts

**Parts of the completion methods**

x

### Teaching (4 cr)

**Type:**

Participation in teaching

**Grading scale:**

0-5

**Evaluation criteria:**

Opintojakson arvosana määräytyy kurssitentin pistemäärän ja laskuharjoitushyvitysten summan perusteella.

**Language:**

Finnish

**Study methods:**

Luennot 38h, 8 harjoituskertaa.

#### Teaching

##### 1/9–3/19/2023 Lectures

##### 3/22–3/22/2023 Exam

x

### Exam (4 cr)

**Type:**

Exam

**Grading scale:**

0-5

**Evaluation criteria:**

Opintojakson arvosana määräytyy lopputentin pistemäärän perusteella. Hyväksyttyyn arvosanaan vaaditaan vähintään puolet maksimipistemäärästä.

**Language:**

English, Finnish