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# Linear Algebra 1

Structure Type: Study unit IST2002 Compulsory / Basic Studies ST 2018 Bachelor of Engineering 1 (2018-2019) Autumn 2 cr Mäkelä, Jarmo Finnish

## Courses During the Academic Year 2018-2019

Impl.Group(s)Study TimeTeacher(s)LanguageEnrolment
4I-ST-1N2018-10-22 – 2018-12-21Jarmo MäkeläFinnish2018-08-20 – 2018-09-17
5I-TT-1N2019-01-07 – 2019-04-10Jarmo MäkeläFinnish2018-12-10 – 2019-01-14

Still need to take the course? See the courses during the academic year 2021-2022.

## Learning Outcomes

Linear algebra is a branch of mathematics which is normally not included in the standard high school curriculum. A notable part of linear algebra concerns the properties of matrices and determinants. In short, matrices are usually expressed as arrays of numbers, where the numbers have been arranged in rows and columns. In a square matrix the numbers of rows and columns are the same, and for every square matrix one may calculate a number, which is known as the determinant of the matrix. Matrices and determinant have various applications, but in this course they are basically applied in the solution of the linear systems of equations. The student learns the basic operations of matrices and determinants, and to solve by means of them linear systems of equations. A concept, which is closely related to matrices, is vector. Vectors are used to describe quantities which, in addition to magnitude, are also associated with direction. In this course the student learns the basics of vector calculus, and to apply vectors when solving simple geometrical problems. In addition, the course involves an extension of basic trigonometry, and provides the basic knowledge on complex numbers and their expressions in the polar form. The last part of the course concerns inequalities.

54 h, which contains 28 h of scheduled contact studies at VAMK and 16 h at the University of Vaasa.

## Contents

Determinants with two rows. The general determinant with n rows. Expansion of the determinant by the given row or column. Basic properties of determinants. Reduction of the determinant. Solution of a linear system of equations by means of determinants (Cramer’s rule). Matrix. The most common types of matrices. Addition of matrices, and the multiplication of a matrix by a number. Matrix multiplication. The inverse of the square matrix. Determination of the inverse of the given matrix by means of Cramer’s rule. expression of a linear system of equations in a matrix form. solution of a linear system of equations by means of matrices. Expansion of basic trigonometry: Trigonometric functions defined by means of the properties of the unit circle. The concept of radian. Basic properties of the trigonometric functions. Expression of the trigonometric function of the given angle as a function of an angle situated in the first quadrant of the unit circle. Sines and cosines of sums and differences of two angles. Since and cosines of double angles. Trigonometric equations and the formulas for their solutions. Graphs of trigonometric functions. The concept of vector. Vector addition and the multiplication of a vector by a number. Three-dimensional Cartesian frame of reference. Vectors represented in a component form. in a Cartesian frame of reference. Vector connecting two points. The length of a vector and the distance between two points. Dot product. The dot product of vectors written in a component form. The angle between two vectors. Cross product and its determination between vectors written in a component form. The connection between cross product and area. Complex numbers and their rules of calculation. Complex plane. Complex numbers written in a polar form. Euler’s formula for the exponential of an imaginary number. Multiplication, division and powering of complex numbers written in a component form. Inequalities: Inequalities of the first and the higher degree, and rational inequalities. Solution of inequalities by means of the sign table.

## Recommended or Required Reading and Other Learning Resources/Tools

S. Alestalo, P. Lehtola, T. Nieminen, A. Rantakaulio: Tekninen matematiikka 1, Tammertekniikka. Material prepared by the teacher.

## Mode of Delivery / Planned Learning Activities and Teaching Methods

Lectures, exercises.

## Assessment Criteria

Grade 5: The student is able to solve problems creatively in almost all the contents of the course.
Grade 3: The student can solve applied problems related with the central contents of the course.
Grade 1: The student can solve basic problems on the central contents of the course.

## Assessment Methods

Homework exercises, assignments, an examination.