Table of Contents
Table of Contents
“Exercises of Matrices and Linear Algebra”
INTRODUCTION
THEORETICAL OUTLINE
EXERCISES
“Exercises of Matrices and Linear Algebra”
“Exercises of Matrices and Linear Algebra”
SIMONE MALACRIDA
In this book, exercises are carried out regarding the following mathematical topics:
matrices and matrix calculus
linear algebra
diagonalization of matrices and canonical bases.
Initial theoretical hints are also presented to make the performance of the exercises understood.
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – THEORETICAL OUTLINE
Matrix definitions
Operations and properties
Matrix calculation
Linear algebra
Diagonalizable matrices and canonical forms
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II – EXERCISES
Exercise 1 _
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercise 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
Exercise 28
Exercise 29
Exercise 30
Exercise 31
INTRODUCTION
INTRODUCTION
In this workbook, some examples of calculation related to matrices and matrix calculus are carried out.
Furthermore, the main theorems used in this discipline are presented.
Matrices are not simply extensions of vectors, but have peculiar properties which make them suitable for the description of complex linear systems and for the characterization of particular topological spaces.
Therefore, their role is central to the development of geometry and advanced mathematical analysis.
In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.
What is presented in this workbook is generally addressed in university-level geometry courses, even if the concept of a matrix is usually already introduced during high school.
I
THEORETICAL OUTLINE
THEORETICAL OUTLINE
Matrix definitions
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A matrix is a table of items sorted by rows and columns .
Given m rows and columns, the matrix is called "m times n" and is denoted by a capital letter.
Each element of the matrix is denoted by two subscripts, the first indicating the row, the second the column.
Vectors can be considered matrices in simplified form, having only one row or one column.
A matrix of dimension 1xn is called a row matrix , if instead it is mx1 it is called a column matrix .
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Operations and properties
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The sum and the difference between matrices having the same dimension is given by the sum of the single elements.
Multiplication by a scalar is done by multiplying each individual element by the scalar.
Multiplication between matrices is carried out in "rows by columns" form and can only be done if the number of columns of the first matrix is equal to the number of rows of the second matrix and we have these formulas for the product :
which is a generalization of the dot product between vectors.
This operation is not commutative, while all other properties of the product and the sum are preserved.
We define 0 as the null matrix made up of zeros only, while the opposite of a matrix is given by the matrix having all the elements multiplied by -1 .
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The following properties hold for the sum:
Property of existence of the neutral element A+0=0+A=A
Property of existence of the opposite element A+(-A)=0
Associative property (A+B)+C=A+(B+C)
Commutative property A+B=B+A
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The following properties hold for the product with a scalar:
Property of existence of the neutral element 1A=A
Associative property (ab)A=a(bA)
Distributive property a(A+B)=aA+aB
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The following properties hold for the product of matrices:
Associative property (AB)C=A(BC)
Distributive property (A+B)C=AC+BC
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The product of matrices also generalizes the product of a matrix and a vector.
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Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 23.04.2023
ISBN: 978-3-7554-4008-6
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