Lecture 28
Andrei Antonenko
April 21, 2003
1
Operators
In this lecture we will start studying the most important part of the course on linear algebra
the theory of operators.
Let V be a vector space. Any linear function from V to V is called the linear o
Lecture 27
Andrei Antonenko
April 14, 2003
1
Orthogonal bases
In this section we will generalize the example from the previous lecture. Let cfw_v1 , v2 , . . . , vn be
an orthogonal basis of the Euclidean space V . Our goal is to nd coordinates of the ve
Lecture 26
Andrei Antonenko
April 11, 2003
1
Orthogonality
Let V be a Euclidean space, and let v and u be 2 vectors in this space. Then we can dene the
angle between these 2 vectors.
Denition 1.1. The angle between two vectors v and u from the vector spac
Lecture 25
Andrei Antonenko
April 7, 2003
1
Euclidean spaces
Denition 1.1. Let V be a vector space. Suppose to any 2 vectors v, u V there assigned a
number from R which will be denoted by v , u such that the following 3 properties hold:
Bilinearity
au1 +
Lecture 24
Andrei Antonenko
April 4, 2003
1
On the row(column) expansion
This lecture we will give a nice example of application of a row expansion to computing the
determinant of a large matrices.
Let An be the matrix with n rows and n columns of the fol
Lecture 23
Andrei Antonenko
April 2, 2003
1
Properties of determinants-2
Now well give a rst motivation of the determinant.
Theorem 1.1 (Criteria of invertibility). A square matrix A is invertible if and only if
det A = 0.
Proof. Lets use elementary row o
Lecture 23 - Addendum
Andrei Antonenko April 2, 2003
1
Proofs of the main results from the lecture
Lets recall the denition from one of the previous lectures. Denition 1.1. Function f (a1 , a2 , . . . , am ) is called multilinear if it is linear in every
Lecture 22
Andrei Antonenko
March 31, 2003
1
Properties of determinants
This lecture we will start studying a properties of determinants, and algorithms of computing
them. Lets recall, that we dened a determinant by the following way:
a11 a12 . . . a1n
a
Lecture 21
Andrei Antonenko
March 28, 2003
1
General properties of area, volume and their generalizations
In this lecture we will give the general denition of the determinant of any square matrix.
On the last lecture we introduced the oriented area of the
Lecture 20
Andrei Antonenko
March 26, 2003
1
Permutations
Let we have n numbers from 1 to n: 1, 2, . . . , n. If we change their order we get their permutation. We will write these numbers in brackets, for example, (34152) is a permutation of
numbers from
Lecture 19
Andrei Antonenko
March 24, 2003
1
Area of the parallelogram
Lets consider a plane R2 . Now we will consider parallelograms on this plane, and compute
their area.
First thing which is clear from elementary geometry is a formula for the area of t
Lecture 18
Andrei Antonenko
March 14, 2003
1
Theoretical facts about image
Now well develop some theory about the image and its dimension and basis.
Let f be a linear function from V to U , and dim V = n. Lets consider the kernel of f . We
can nd the basi
Lecture 17
Andrei Antonenko
March 12, 2003
1
Kernel: its dimension and basis
Last lecture we saw that the kernel of a linear function is a vector space. Each vector space
has a dimension and basis this lecture well try to determine them for the kernel.
Le
Lecture 16
Andrei Antonenko
March 10, 2003
1
Image and kernel
Last lecture we studied image and kernel of a linear function. Now we will prove one of the
properties of image and kernel. First lets consider kernel.
Let f : V U be a linear function, and let
Lecture 15
Andrei Antonenko
March 07, 2003
1
Homogeneous systems
Last time we studied linear functions. To continue our theory about them we have to study
another topic, which we will use in our future lectures.
Recall, that the system is called homogeneo
Lecture 14
Andrei Antonenko
March 05, 2003
1
Functions
In previous lectures we worked with algebraic structures sets with operations dened on
them. Now we will consider another important thing in mathematics functions.
Let A and B be 2 sets. Function f fr
Lecture 13
Andrei Antonenko
March 3, 2003
1
Dimension and basis of the span
Last lecture we formulated the problem of nding the basis and the dimension of the span of
given vectors. This lecture we will give the algorithm to determine these characteristic
Lecture 12
Andrei Antonenko
February 28, 2003
1
Examples of bases and dimensions
Last lecture we stated the result that each basis has the same number of vectors. From this
result very important corollary follows.
Corollary 1.1. If the dimension of the ve
Lecture 11
Andrei Antonenko
February 26, 2003
1
Examples of bases
Last time we studied bases of vector spaces. Today were going to give some examples of bases.
Example 1.1. Consider the vector space P2 the space of polynomials with degree less than
or equ
Lecture 10
Andrei Antonenko
February 24, 2003
1
Meaning of linear dependence and independence
On the last lecture we stated the result that if the system of vectors is linearly dependent, then
at least one vector of the. system can be expressed as a linea
Lecture 9
Andrei Antonenko
February 21, 2003
1
Linear combinations
Denition 1.1. Let V be a vector space. A vector v V is a linear combination of vectors
u1 , u2 , . . . , un if there exist a1 , a2 , . . . , an k such that
v = a1 u1 + a2 u2 + + an un .
(1
Lecture 8
Andrei Antonenko
February 19, 2003
1
Matrix equations and the inverse
1.1
Discussion of the algorithm - Part 2
Last time we proved the following result:
Lemma 1.1.
1. If the square matrix A is invertible, then its RREF is the identity matrix.
2.
Lecture 7
Andrei Antonenko
February 14, 2003
1
Matrix equations and the inverse
1.1
Algorithm of solving
Suppose we have 2 given matrices A and B , and wed like to nd a matrix X such that
AX = B.
(1)
Note, that B should not be a square matrix, for example
Lecture 6
Andrei Antonenko
February 12, 2003
1
Multiplicative inverse
In this lecture we will continue with the properties of matrix operations.
(M4) Existence of the multiplicative inverse. Matrix B is called the inverse for the
square matrix A if BA = A
Lecture 5
Andrei Antonenko
February 10, 2003
1
Matrices
Now well start studying new algebraic object matrices.
Denition 1.1. The matrix is a (rectangular) table of the elements of R. (Actually, we can
consider matrices over elds other than R in the future
Lecture 4
Andrei Antonenko
February 05, 2003
1
Analysis of the system
Now, we can formulate the main theoretical result about the system. This theoretical result
follows directly from our practical method of solving them.
Case 1 If during our procedures w
Lecture 3
Andrei Antonenko
February 03, 2003
1
Linear systems and their solutions
This lecture were going to speak about the most important and boring part of linear algebra
about general linear systems we will learn how to solve and analyze them.
Deniti
Lecture 2
Andrei Antonenko
January 31, 2003
1
Introduction to linear equations
Last lecture we were talking about the general mathematical concepts, like a concept of a
number, a concept of a set, a concept of an operation. This lecture we will start stud
Lecture 1
Andrei Antonenko
January 30, 2003
1
Introduction
The rst mathematical object which every person meets even in the childhood is a number.
So, the number is often considered as a main mathematical object, which is not really true
(though, it is no
AMS210.01. Quiz 2. Solutions.
Andrei Antonenko
1. Which of the following statements are correct? If yes, provide the explanation. If no, provide the counterexample.
(a) Let A, B, C be square matrices of the same size. Then if AB = AC , then B = C .
Soluti