L. 2-1 Basic concepts of vector algebra. Linear operations over vectors.

Baseline decomposition.

Basic concepts vector algebra

The vector is called a set of all directed segments having the same length and direction
.

Properties:

Linear operations over vectors

1.

Rule parallelogram:

FROM umbletwo vectors and called vector coming out of their general principle and being a diagonal of a parallelogram-ma built in vectors and as on the sides.

Polygon rule:

To build the amount of any number of vectors, it is necessary to put the beginning of the 2nd 3rd sample vector at the end of the 2nd, at the end of the 2nd - beginning of 3rd, etc. The vector closing the resulting broken line is the sum. Its beginning coincides with the beginning of the 1st, and the end - with the end of the latter.

Properties:

2.

Work vector number , called vector satisfying conditions:
.

Properties:

3.

Differencevectors and call vector equal and vector opposite vector .
.

- The law of the opposite element (vector).

Baseline decomposition

The sum of vectors is determined by the only way.
(only ). The inverse operation is the decomposition of the vector into several components, ambiguous:. In order to make it unambiguous, you must specify the directions by the decomposition of the vector under consideration, or, as they say, you must specify basis.

When determining the basis, it is essential to the requirement of noncomplaunarity and non-salinearity of vectors. To understand the meaning of this requirement, it is necessary to consider the concept of linear dependence and linear independence of vectors.

Arbitrary expression of the form:, called linear combinationvectors
.

Linear combination of several vectors is called trivialif all its coefficients are zero.

Vectors
called linearly dependentif there is a non-trivial linear combination of these vectors equal to zero:
(1), provided
. If equality (1) takes place only at all
at the same time equal zero, then nonzero vectors
will be linearly independent.

Easy to prove any two collinear vectors are linearly dependent, and two non-rigid vector is linearly independent.

Proof Let's start with the first approval.

Let vectors and collinear. We show that they are linearly dependent. Indeed, if they are collinear, they differ from each other only on a numerical factor, i.e.
, hence
. Since the resulting linear combination is clearly nontrivial and equal to "0", then vectors and linearly dependent.

Consider now two non-halinar vectors and . We prove that they are linearly independent. Proof construct from nasty.

Suppose they are linearly dependent. Then there must be a nontrivial linear combination
. Let's pretend that
, then
. The resulting equality means that vectors and collinear, contrary to our initial assumption.

Similarly, you can prove: any three compartment vectors are linearly dependent, and two noncomplanar vector are linearly independent.

Returning to the concept of the basis and the task of the decomposition of the vector in a specific basis, it can be said that the basis on the plane and in space is formed from the set of linearly independent vectors.Such a concept of basis is common, because It applies to the space of any number of measurements.

Expression of the form:
, is called the decomposition of the vector by vectors ,…,.

If we consider the basis in three-dimensional space, then the decomposition of the vector basisus
will be
where
-coordinates of the vector.

The task of expanding an arbitrary vector in some basis is very important to the following statement: any vectormay be the only way unfolded in this base
. In other words, coordinates
for any vector regarding Basisa
definitively.

The introduction of the basis in space and on the plane allows you to put in line with each vector an ordered triple (pair) numbers is its coordinates. This very important result that allows you to establish a link between geometric objects and numbers, makes it possible to analytically describe and explore the position and movement of physical objects.

The totality of the point and base is called Coordinate system.

If the vectors forming the basis are single and pairly perpendicular, then the coordinate system is called rectangulara Base orthonormal.

L. 2-2 works of vectors

Baseline decomposition

Consider vector
as defined by its coordinates:
.

- Components of the vector in the directions of basic vectors
.

Expression of type
called decomposition of vector basisus
.

Similarly, you can decompose basisus
vector
:

.

Cosine Corners formed by the Vector with basic ortes
called guides cosinees

;
;
.

Scalar product of vectors.

Scalar product of two vectors and called the number equal to the product of the modules of these vectors on the cosine of the corner between them

The scalar product of two vectors can be viewed as a product of a module of one of these vectors on the orthogonal projection of another vector to the direction of the first
.

Properties:

If the coordinates of the vectors are known
and
, after performing the expansion of the vectors of the basis
:
and
, Find

because
,
T.

.

.

The condition perpendicularity of vectors:
.

Conditions of collinearity rectors:
.

Vector artwork vectors

or

Vector product vector on vector called such vector
which satisfies the conditions:

Properties:

The considered algebraic properties make it possible to find an analytical expression for vector art through the coordinates of the components of the vectors in the orthonormal basis.

Given:
and
.

because .
,
,
,
,
,
T.

. This formula can be recorded in short, in the form of a third-order determinant:

.

Mixed vectors

Mixed product of three vectors ,and called a number equal to vector work
multiplied scalar to vector .

True the following equality:
, so mixed work is recorded
.

As follows from the definition, the result of a mixed product of three vectors is the number. This number has a visual geometric meaning:

Module of mixed work
equal to the amount of parallelepiped, built on the above general to the beginning vectors ,and .

Properties of mixed work:

If vectors ,,specified in the orthonormal basis
its coordinates, the calculation of the mixed work is carried out by the formula

.

Indeed, if
T.

;
;
, then
.

If vectors ,,complian, then vector work
perpendicular to the vector . And vice versa if
The volume of parallelepiped is zero, and this is only possible when the compartment vectors (linearly dependent).

Thus, three vectors are compartment, then and only if their mixed work is zero.

Linear dependence and linear independence of vectors.
Basis vectors. Affine coordinate system

The audience has a trolley with chocolates, and every visitor today will get a sweet couple - analytical geometry with a linear algebra. Two sections will be touched upon this article. higher MathematicsAnd we will see how they get along in one wrapper. Make a pause, be boring "Twix"! ... damn, well, nonsense spore. Although okay, I will not score, in the end, there should be a positive attitude to study.

Linear dependence of vectors, linear independence vectors, basis vectors and others. Terms have not only geometric interpretation, but, above all, algebraic meaning. The very concept of "vector" in terms of linear algebra is not always the "ordinary" vector, which we can portray on the plane or in space. It is not necessary to go far for evidence, try to draw a vector of five-dimensional space. . Or the weather vector, followed by which I just went to Gismeteo: - Temperature and atmosphere pressure respectively. An example, of course, incorrect from the point of view of the properties of vector space, but, nevertheless, no one prevents formalize these parameters by vector. Respiratory of autumn ...

No, I'm not going to ship you theory, linear vector spaces, the task is to understand Definitions and theorems. New terms (linear dependence, independence, linear combination, basis, etc.) are applicable to all vectors from an algebraic point of view, but examples will be given geometric. Thus, everything is simple, accessible and visual. In addition to the tasks of analytical geometry, we will look at and some typical tasks of algebra. To master the material, it is advisable to get acquainted with the lessons Vectors for teapots and How to calculate the determinant?

Linear dependence and independence of plane vectors.
Plane base and affinity coordinate system

Consider the plane of your computer table. (just a table, bedside tables, floor, ceiling, who likes what). The task will consist in the following actions:

1) Select base plane. Roughly speaking, the countertops have a length and width, so it is intuitive that two vectors will be required to build a base. One vector is clearly not enough, three vectors - Lishka.

2) Based on the selected basis set the coordinate system (coordinate grid) to assign coordinates to all subjects on the table.

Do not be surprised, first the explanations will be on your fingers. And, on yours. Please place finger of the left hand On the edge of the table top so that he looked into the monitor. It will be vector. Now place little finger right hand On the edge of the table is exactly the same - that it is directed to the monitor screen. It will be vector. Smile, you wonderfully look! What can be said about the vectors? These vectors collinearnyand therefore linelo expressed in each other:
Well, or vice versa:, where - some number other than zero.

Picture of this action can be viewed at the lesson Vectors for teapotswhere I explained the vector multiplication rule for a number.

Will your fingers set the basis on the computer table plane? Obviously, no. Collinear vectors travel there and here one Direction, and the plane has a length and width.

Such vectors are called linearly dependent.

Reference: The words "linear", "linearly" denote the fact that there are no squares in mathematical equations, expressions, cubes, other degrees, logarithms, sinuses, etc. There are only linear (1st degree) of expressions and dependencies.

Two vector planes linearly dependent Then and only when they are collinear.

Cross your fingers on the table to be between them any angle, except 0 or 180 degrees. Two vector planeslinelo notdependent in that and only if they are not collinear. So, the basis is obtained. It is not necessary to embarrass that the basis turned out to be "oblique" with imperpendicular vectors of different lengths. Very soon we will see that for its construction is not only an angle of 90 degrees, and not only single, equal length vectors

Any Vector plane the only way Disclosed by base:
where - valid numbers. Numbers are called coordinates of the vector In this base.

Also say that vector Posted in the form linear combination Basic vectors. That is, the expression is called decomposition of vectorbasisus or linear combination Basic vectors.

For example, we can say that the vector is decomposed on the orthonormal basis of the plane, and it can be said that it is represented as a linear combination of vectors.

Formulate definition of Basisa formally: Base plane called a pair of linear independent (nonollylinear) vectors, , wherein any The vector of the plane is a linear combination of basic vectors.

The essential point of definition is the fact that the vectors are taken in a certain order. Bases - These are two completely different bases! As the saying goes, the little finger of the left hand will not rearrange the Mizinza right hand.

The basis figured out, but it is not enough to set the coordinate grid and assign the coordinates to each object of your computer table. Why not enough? Vectors are free and wandered throughout the plane. So how to assign the coordinates of those small dirty points of the table, which remained after a rapid weekend? We need a starting reference. And such a guideline is a familiar point - the beginning of the coordinates. We understand the coordinate system:

I will start with the "school" system. Already at the introductory lesson Vectors for teapots I highlighted some differences between the rectangular coordinate system and the orthonormal basis. Here is the standard picture:

When they say O. rectangular coordinate system, most often they mean the origin of the coordinates, coordinate axes and scale along the axes. Try to dial in the search engine "Rectangular coordinate system", and you will see that many sources will tell you about familiar with the 5-6th class coordinate axes and how to postpone points on the plane.

On the other hand, it seems that the rectangular coordinate system can be determined through an orthonormal basis. And it is almost like that. The wording sounds as follows:

The beginning of the coordinates, I. Ortonormalbasis set cartesian Rectangular Plane Coordinate System . That is, a rectangular coordinate system definite Determined by the only point and two single orthogonal vectors. That is why you see the drawing that I was led above - in geometric tasks, often (but not always) draw vectors, and coordinate axes.

I think everyone is clear that with the help of a point (start of coordinates) and the orthonormal basis Any point of the plane and any plane vectoryou can assign coordinates. Figuratively speaking, "on the plane everything can be numbered."

Are the coordinate vectors obliged to be isolated? No, they can have an arbitrary non-zero length. Consider the point and two orthogonal vectors of arbitrary nonzero length:

Such a basis is called orthogonal. The origin of the coordinates with vectors set the coordinate grid, and any point of the plane, any vector has its own coordinates in this base. For example, or. The obvious inconvenience is that the coordinate vectors in general Have different lengths other than one. If the lengths are equal to one, then the usual orthonormal basis is obtained.

! Note : In the orthogonal basis, as well as below in affine bases of the plane and space, the units on the axes are considered Conditional. For example, in one unit along the abscissa axis, it contains 4 cm, in one unit along the ordinate axis 2, see this information is enough to translate the "non-standard" coordinates to "our ordinary centimeters" if necessary.

And the second question for which the answer is already given - is it necessary to equal 9 degrees between basic vectors? Not! As the definition says, basic vectors must be only nonollylinear. Accordingly, the angle can be anyone except 0 and 180 degrees.

Point plane called The beginning of the coordinates, I. nonollylinear vectors , ask affine coordinate plane system :

Sometimes such a coordinate system is called kosholnaya system. As examples in the drawing, dots and vectors are depicted:

As you understand, the affine coordinate system is even less convenient, it does not work formulas for the vectors and segments that we considered in the second part of the lesson Vectors for teapotsMany tasty formulas related to scalar product vectors. But there are valid rules for the addition of vectors and multiplication of the vector by the number, the segment division formula in this regard, as well as some more tasks that we will consider soon.

And the conclusion that the most convenient private case of the affinity coordinate system is the decartian rectangular system. Therefore, it, native, most often and has to be contemplating. ... However, everything in this life is relative - there are a lot of situations in which the Kosholnaya is appropriate (or what other, for example, polar) coordinate system. Yes, and humanoids such systems may come to taste \u003d)

Go to the practical part. All tasks of this lesson are valid for both a rectangular coordinate system and for a common affional case. There is nothing difficult here, all the material is even available to a schoolboy.

How to determine the collinearity of the plane vectors?

Typical thing. In order for two plane vector were collinear, it is necessary and enough so that their relevant coordinates are proportional to. According to the creature, this is the redeeble detailing of the obvious relationship.

Example 1.

a) check whether Collinearny vectors .
b) whether the basis forms the vectors ?

Decision:
a) find out if there is a vectors The proportionality coefficient, such to be carried out by equality:

I will definitely tell about the "Pzhonskaya" species of the application of this rule, which is quite rolling in practice. The idea is to immediately make a proportion and see if it will be true:

Make a proportion from the relationship of the corresponding coordinates of the vectors:

Redfish:
Thus, the corresponding coordinates are proportional, therefore,

The attitude could be converted on the contrary, it is an equal version:

For self-test, it is possible to use the fact that the collinear vectors are linearly expressed in each other. In this case, there are equality . Their justice is easily checked through elementary actions with vectors:

b) Two plane vector form a basis, if they are not collinear (linearly independent). Explore the collinearity vectors . Make a system:

From the first equation it follows that, from the second equation it follows that, it means that the system is incomplete (No solutions). Thus, the corresponding coordinates of the vectors are not proportional.

Output: Vectors are linearly independent and form a basis.

The simplified version of the solution looks like this:

Make a proportion from the corresponding vectors coordinates :
It means that these vectors are linearly independent and form a basis.

Usually, this option is not marked by reviewers, but the problem arises in cases where some coordinates are zero. Like this: . Or so: . Or so: . How to act through the proportion? (Indeed, it is impossible to share for zero). It was for this reason that I called the simplified decision "Pzhonsky".

Answer:a), b) form.

A small creative example for self-decide:

Example 2.

With what value of the parameter vector Will collinearins?

In the sample solution, the parameter is found through proportion.

There is an elegant algebraic method of checking vectors for collinearity., We systematize our knowledge and fifth item just add it:

The following statements are equivalent for two plane vectors.:

2) vectors form basis;
3) vectors are not collinear;

+ 5) The determinant composed of the coordinates of these vectors is different from zero.

Respectively, the following opposite statements are equivalent.:
1) vectors are linearly dependent;
2) vectors do not form the basis;
3) collinear vectors;
4) vectors can be linearly expressed in each other;
+ 5) The determinant composed of the coordinates of these vectors is zero.

I really and very hope that this moment All those who meet the terms and approval are already understood.

Consider a new one, the fifth point: two vector planes Collinearny then and only if the determinant composed of the data coordinates of the vectors is zero:. To apply this feature, naturally, you need to be able to find identifies.

Decisive Example 1 second way:

a) Calculate the determinant composed of the coordinates of the vectors :
So, these collinear vectors.

b) Two plane vector form a basis, if they are not collinear (linearly independent). Calculate the determinant composed of the coordinates of vectors :
So, vectors are linearly independent and form a basis.

Answer:a), b) form.

It looks much more compact and prettier than the solution with proportions.

With the help of the considered material, not only the collinearity of vectors can be installed, but also to prove the parallelism of segments, direct. Consider a pair of tasks with specific geometric shapes.

Example 3.

Dana vertices of a quadricle. Prove that the quadril is a parallelogram.

Evidence: The drawing is not necessary in the task, since the solution will be purely analytical. Remember the definition of the parallelogram:
Parallelogram Called a quadricle, who has opposite sides pairly parallel.

Thus, it is necessary to prove:
1) parallelism of opposite sides and;
2) parallelism of opposite sides and.

We prove:

1) Find vectors:

2) Find vectors:

It turned out the same vector ("on school" - equal vectors). Collinearity is completely obvious, but it is better to make a decision with an alignment. Calculate the determinant composed of the coordinates of the vectors:
It means that these are collinear vectors, and.

Output: The opposite sides of the quadril is parallel parallel, it means that it is a parallelogram by definition. Q.E.D.

More figures Good and different:

Example 4.

Dana vertices of a quadricle. Prove that the quadril is a trapezium.

For a more strict wording of the proof, it is better, of course, to get the definition of a trapezoid, but it is enough and just remember how it looks like.

This is a task for an independent solution. Complete solution at the end of the lesson.

And now it's time to quietly move out of the plane into space:

How to determine the collinearity of space vectors?

The rule is very similar. In order for two vessel vectors to be collinear, it is necessary and enough for their respective coordinates to be proportional to.

Example 5.

Find out whether the collinear will be the following vectors of space:

but) ;
b)
in)

Decision:
a) Check whether there is a ratio of proportionality for the corresponding coordinates of the vectors:

The system has no solution, it means that the vectors are not collinear.

"Simplified" is issued by checking the proportion. In this case:
- The relevant coordinates are not proportional, it means that the vectors are not collinear.

Answer: Vectors are not collinear.

b-c) These are items for an independent decision. Try it to arrange in two ways.

There is a method for checking spatial vectors to collinearity and through the third order determinant, this method LED in the article Vector artwork vectors.

Similar to the flat case, the considered toolkit can be used to study the parallelism of spatial segments and direct.

Welcome to the second section:

Linear dependence and independence of vectors of three-dimensional space.
Spatial Base and Affine Coordinate System

Many of the laws that we looked at the plane will be fair for space. I tried to minimize the abstract on the theory, since the lion's share of information is already degraded. However, I recommend to read the introductory part carefully, as new terms and concepts will appear.

Now instead of the plane of the computer table, we examine three-dimensional space. First create its basis. Someone is now located in the room, someone on the street, but in any case, we can not go anywhere from three dimensions: widths, lengths and heights. Therefore, three spatial vectors will be required to build a base. One or two vectors are small, the fourth is superfluous.

And again breathe on the fingers. Please raise your hand up and pop up in different sides big, index and middle finger . It will be vectors, they look at different directions, have a different length and have different angles between themselves. Congratulations, the basis of three-dimensional space is ready! By the way, it is not necessary to demonstrate such teachers, no matter how cool your fingers, and the definitions are not going anywhere \u003d)

Next, asked an important issue, any three vectors form the basis of three-dimensional space? Please press the three fingers tightly to the computer table tabletop. What happened? Three vectors are located in the same plane, and, roughly speaking, we lost one of the measurements - the height. Such vectors are complinary And, it is quite obvious that the basis of three-dimensional space does not create.

It should be noted that the compartment vectors are not required to lie in the same plane, they can be in parallel planes (just do not do it with your fingers, so only Salvador gave \u003d)).

Definition: Vectors are called complinaryIf there is a plane with which they are parallel. It is logical here that if such a plane does not exist, then the vectors will not compartment.

Three compartment vectors are always linearly dependent., that is, linearly expressed in each other. For simplicity, we will again imagine that they lie in the same plane. First, the vectors are not enough that the companaries may also be collinear, then any vector can be expressed through any vector. In the second case, if, for example, vectors are not collinear, then the third vector is expressed through them the only way: (And why - easy to guess based on the materials of the previous section).

Fairly reverse statement: Three noncomplete vectors are always linearly independent, that is, in no way expressed in one friend. And, obviously, only such vectors can form a three-dimensional basis.

Definition: The basis of three-dimensional space called a tripler linearly independent (noncomplete) vectors, taught, with any vector space the only way Disclosed on this basis, where - the coordinates of the vector in this base

I remind you can also say that the vector is presented in the form of linear combination Basic vectors.

The concept of the coordinate system is introduced in the same way as for a flat case, just one point and any three linearly independent vectors:

The beginning of the coordinates, I. noncomplenar vectors taken in a definite, ask affine coordinate system of three-dimensional space :

Of course, the coordinate mesh "oblique" and poorly turning, but, nevertheless, the constructed coordinate system allows us definite Determine the coordinates of any vector and coordinates of any point of space. Similarly, the plane in the affine coordinate system will not work for some formulas that I have already mentioned.

The most familiar and convenient private case of an affine coordinate system, how everyone is guessing is rectangular Space Coordinates System:

Point space called The beginning of the coordinates, I. Ortonormalbasis set cartepow rectangular space coordinate system . Familiar picture:

Before moving to practical tasks, we again systematize the information:

For three vectors of space are equivalent to the following statements:
1) vectors are linearly independent;
2) vectors form basis;
3) vectors are not compartment;
4) vectors can not linearly express each other;
5) The determinant composed of the coordinates of these vectors is different from zero.

Opposite statements, I think, are understandable.

Linear dependence / independence of space vectors is traditionally checked using the determinant (paragraph 5). The remaining practical tasks Will wearing a pronounced algebraic character. It's time to hang on a nail geometric club and wrapping a baseball bat linear algebra:

Three vector vectors Compliannas then and only if the determinant drawn up from the coordinates of these vectors is zero: .

I draw attention to a small technical nuance: the coordinates of the vectors can be recorded not only in the columns, but also in the string (the value of the determinant will not change from this - see the properties of the determinants). But much better in columns, since it is more profitable for solving some practical tasks.

Thus, the readers who are a little challenged methods for calculating the determinants, and may generally be focused on them, I recommend one of my oldest lessons: How to calculate the determinant?

Example 6.

Check whether the three-dimensional basis form the following vectors:

Decision: In fact, the whole decision is reduced to the calculation of the determinant.

a) Calculate the determinant composed of the coordinates of the vectors (the determinant is disclosed on the first line):

It means that the vectors are linearly independent (not compartment) and form the basis of three-dimensional space.

Answer: These vectors form a basis

b) this item for an independent solution. Complete solution and answer at the end of the lesson.

Creative tasks are found:

Example 7.

With what value of the vector parameter will be compartment?

Decision: The vectors are compartment if and only if the determinant drawn up from the data coordinates of the vectors is zero:

Essentially, it is required to solve the equation with the determinant. We turn to zeros as the kerchings on the tubes - the determinant is most advantageous to disclose the second line and immediately get rid of the minuses:

We carry out further simplifications and reduce the simplest linear equation:

Answer: for

It is easy to perform a check, for this you need to substitute the received value to the original determinant and make sure that , Overlook it again.

In conclusion, consider another one typical problemwhich wears more algebraic and traditionally turns on to a linear algebra. It is so common that deserves a separate topic:

Prove that 3 vectors form a three-dimensional basis
and find the coordinates of the 4th vector in this basis

Example 8.

Vast vectors. Show that vectors form the basis of three-dimensional space and find the coordinates of the vector in this base.

Decision: First we disassemble with the condition. For the condition, four vectors are given, and, as you can see, they already have coordinates in some basis. What a basis is not interested in us. And you are interested in the following thing: three vectors may well form a new basis. And the first stage completely coincides with the solution of Example 6, it is necessary to check whether the vectors are really linearly independent:

Calculate the determinant composed of the coordinates of the vectors:

So, the vectors are linearly independent and form the basis of three-dimensional space.

! Important : Coordinates of vectors before Record in columns determinant, and not in the string. Otherwise there will be a confusion in the further solution algorithm.

Basis (Dr. Greek. βασις, base) - a set of such vectors in the vector space, that any vector of this space may be the same in the form of a linear combination of vectors from this set - basic vectors

The basis in the space R n is called any system from n.- Linely independent vectors. Each vector of R n, which are not included in the basis, can be represented as a linear combination of basic vectors, i.e. Dispatch by base.
Let be the basis of the space R n and. Then there are such numbers λ 1, λ 2, ..., λ n, which .
The decomposition coefficients λ 1, λ 2, ..., λ n are called vector coordinates in the base. If the basis is set, the vector coefficients are definitely determined.

Comment. In each n.-Herous vector space you can choose countless different bases. In various bases, the same vector has different coordinates, but the only base in the selected basis. Example. Decay the vector by base.
Decision. . We substitute the coordinates of all vectors and perform actions on them:

Equating the coordinates, we obtain the system of equations:

I solve it: .
Thus, we get a decomposition: .
In the base, the vector has coordinates.

End of work -

This topic belongs to the section:

Vector concept. Linear operations over vectors

The vector is called a directional segment having a certain length of a certain length of which one of the limiting points. The length of the vector is called its module and is indicated by the vector module. The vector is called zero denotes if the beginning and end of it coincides with the zero vector does not have a certain ..

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