Expansion of an arbitrary vector in terms of the basis. Expansion of a vector in basis

Basis(Old Greek βασις, base) is a set of vectors in a vector space such that any vector of this space can be uniquely represented as a linear combination of vectors from this set - basis vectors

A basis in the space R n is any system from n-linearly independent vectors. Each vector from R n not included in the basis can be represented as a linear combination of basis vectors, i.e. expand on the basis.
Let be a basis of the space R n and. Then there are numbers λ 1, λ 2, ..., λ n such that .
The expansion coefficients λ 1, λ 2,…, λ n, are called the coordinates of the vector in the basis B. If the basis is given, then the coefficients of the vector are determined uniquely.

Comment. In each n-dimensional vector space, you can choose countless different bases. In different bases, the same vector has different coordinates, but the only ones in the chosen basis. Example. Expand the vector in basis.
Solution. ... Substitute the coordinates of all vectors and perform actions on them:

Equating the coordinates, we get the system of equations:

Let's solve it: .
Thus, we get the decomposition: .
In the basis, the vector has coordinates.

End of work -

This topic belongs to the section:

Vector concept. Linear operations on vectors

A vector is a directed segment having a certain length, ie a segment of a certain length, which has one of its bounding points .. the length of a vector is called its modulus and is denoted by the vector modulus symbol .. a vector is called zero is denoted if its beginning and end coincide, the zero vector has no definite ..

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

Expansion of a vector in basis.

Basic concepts of vector algebra

A vector is the set of all directed segments having the same length and direction
.

Properties:

Linear operations on vectors

1.

Parallelogram rule:

WITH ummah two vectors and is called a vector originating from their common origin and being the diagonal of the parallelogram built on the vectors and both on the sides.

Polygon rule:

To construct the sum of any number of vectors, you need to put the beginning of the 2nd at the end of the 1st term of the vector, the beginning of the 3rd at the end of the 2nd, etc. The vector that closes the resulting polyline is the sum. Its beginning coincides with the beginning of the 1st, and the end - with the end of the latter.

Properties:

2.

Product of vector by the number , is called a vector satisfying the conditions:
.

Properties:

3.

Difference vectors and called the vector equal to the sum of the vector and the vector opposite to the vector , i.e.
.

- the law of the opposite element (vector).

Expansion of a vector in basis

The sum of vectors is determined in a unique way
(but only ). The reverse operation - the decomposition of the vector into several components, is ambiguous: In order to make it unambiguous, it is necessary to indicate the directions in which the expansion of the vector under consideration occurs, or, as they say, it is necessary to indicate basis.

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

An arbitrary expression of the form:, called linear combination vectors
.

A linear combination of several vectors is called trivial if all its coefficients are equal to zero.

Vectors
are called linearly dependent if there is a nontrivial linear combination of these vectors equal to zero:
(1), provided
... If equality (1) holds only for all
simultaneously equal to zero, then the nonzero vectors
will be linearly independent.

It is easy to prove: any two collinear vectors are linearly dependent and two non-collinear vectors are linearly independent.

We start the proof with the first statement.

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

Consider now two non-collinear vectors and ... Let us prove that they are linearly independent. We construct the proof by contradiction.

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

Similarly, one can prove: any three coplanar vectors are linearly dependent and two noncoplanar vectors are linearly independent.

Returning to the concept of a basis and to the problem of vector expansion in a certain basis, we can say that the basis on the plane and in space is formed from a set of linearly independent vectors. This concept of a basis is general, since it applies to a space of any number of dimensions.

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

If we consider a basis in three-dimensional space, then the expansion of the vector on the basis
will
, where
-vector coordinates.

In the problem of expanding an arbitrary vector in some basis, the following statement is very important: any vectorcan be uniquely expanded in a given basis
. In other words, the coordinates
for any vector on the basis
is uniquely determined.

The introduction of a basis in space and on a plane allows us to associate each vector an ordered triple (pair) of numbers - its coordinates. This very important result, which makes it possible to establish a connection between geometric objects and numbers, makes it possible to analytically describe and investigate the position and movement of physical objects.

The collection of a point and a basis is called coordinate system.

If the vectors forming the basis are unit and pairwise perpendicular, then the coordinate system is called rectangular, and the basis orthonormal.

L. 2-2 Product of vectors

Expansion of a vector in basis

Consider a vector
given by its coordinates:
.

- vector components along the directions of the basis vectors
.

Expression of the form
is called the decomposition of the vector on the basis
.

Similarly, you can decompose on the basis
vector
:

.

The cosines of the angles formed by the vector under consideration with base units
are called direction cosines

;
;
.

Dot product of vectors.

Dot product of two vectors and is the number equal to the product of the moduli of these vectors by the cosine of the angle between them

The scalar product of two vectors can be viewed as the product of the modulus of one of these vectors by the orthogonal projection of the other vector by the direction of the first
.

Properties:

If the coordinates of the vectors are known
and
, then, after performing the decomposition of vectors in the basis
:
and
, we will find

since
,
, then

.

.

The condition of vectors perpendicularity:
.

Rectors collinearity condition:
.

Vector product of vectors

or

By vector product by vector per vector such a vector is called
that satisfies the conditions:

Properties:

The considered algebraic properties make it possible to find an analytical expression for the vector product in terms of the coordinates of the component vectors in the orthonormal basis.

Given:
and
.

since ,
,
,
,
,
,
, then

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

.

Mixed product of vectors

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

The following equality is true:
, so the mixed product is written
.

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

Mixed work module
is equal to the volume of the parallelepiped built on reduced to common origin vectors ,and .

Mixed work properties:

If vectors ,,are given in an orthonormal basis
by its coordinates, the calculation of the mixed product is carried out according to the formula

.

Indeed, if
, then

;
;
, then
.

If vectors ,,are coplanar, then the cross product
perpendicular to vector ... Conversely, if
, then the volume of the parallelepiped is zero, and this is possible only if the vectors are coplanar (linearly dependent).

Thus, three vectors are coplanar if and only if their mixed product is zero.

The basis of space is called a system of vectors in which all other vectors of the space can be represented as a linear combination of vectors included in the basis.
In practice, this is all quite simple to implement. The basis, as a rule, is checked on the plane or in space, and for this it is necessary to find the determinant of the matrix of the second, third order composed of the coordinates of the vectors. Below are schematically written conditions under which vectors form a basis

To expand vector b in basis vectors
e, e ..., e [n], it is necessary to find the coefficients x, ..., x [n] for which the linear combination of the vectors e, e ..., e [n] is equal to the vector b:
x1 * e + ... + x [n] * e [n] = b.
For this, the vector equation should be transformed to the system linear equations and find solutions. It's also fairly straightforward to implement.
The found coefficients x, ..., x [n] are called coordinates of the vector b in the basis e, e ..., e [n].
Let's move on to practical side themes.

Decomposition of a vector in terms of basis vectors

Objective 1. Check if vectors a1, a2 form a basis on the plane

1) a1 (3; 5), a2 (4; 2)
Solution: We compose the determinant from the coordinates of the vectors and calculate it

The determinant is not zero, hence vectors are linearly independent, and therefore form a basis.

2) a1 (2; -3), a2 (5; -1)
Solution: Calculate the determinant composed of vectors

The determinant is equal to 13 (not equal to zero) - from this it follows that the vectors a1, a2 are a basis on the plane.

---=================---

Let's consider typical examples from the IAPM program in the "Higher Mathematics" discipline.

Objective 2. Show that the vectors a1, a2, a3 form a basis of a three-dimensional vector space, and expand the vector b in this basis (when solving a system of linear algebraic equations use Cramer's method).
1) a1 (3; 1; 5), a2 (3; 2; 8), a3 (0; 1; 2), b (−3; 1; 2).
Solution: First, consider the system of vectors a1, a2, a3 and check the determinant of the matrix A

built on vectors other than zero. The matrix contains one zero element, so it is more expedient to calculate the determinant as a schedule for the first column or the third row.

As a result of the calculations, we found that the determinant is nonzero, therefore vectors a1, a2, a3 are linearly independent.
By definition, vectors form a basis in R3. Let us write down the schedule of the vector b in the basis

Vectors are equal when their corresponding coordinates are equal.
Therefore, from the vector equation we obtain the system of linear equations

Let's solve the SLAE Cramer's method... For this, we write the system of equations in the form

The main determinant of the SLAE is always equal to the determinant composed of the basis vectors

Therefore, in practice, it is not counted twice. To find auxiliary determinants, we put a column of free members in place of each column of the main determinant. Determinants are calculated by the rule of triangles



Substitute the found determinants into Cramer's formula



So, the expansion of the vector b in the basis has the form b = -4a1 + 3a2-a3. The coordinates of the vector b in the basis a1, a2, a3 will be (-4,3,1).

2)a1 (1; -5; 2), a2 (2; 3; 0), a3 (1; -1; 1), b (3; 5; 1).
Solution: Check the vectors for a basis - compose the determinant from the coordinates of the vectors and calculate it

The determinant is not equal to zero, therefore vectors form a basis in space... It remains to find the schedule of the vector b through the given basis. For this, we write down the vector equation

and transform to the system of linear equations

We write down matrix equation

Further, for Cramer's formulas, we find auxiliary determinants



Applying Cramer's formulas



So the given vector b has a schedule through two vectors of the basis b = -2a1 + 5a3, and its coordinates in the basis are equal to b (-2,0, 5).

In vector calculus and its applications, the decomposition problem is of great importance, which consists in representing a given vector as a sum of several vectors, called components of a given

vector. This problem, which in the general case has an innumerable set of solutions, becomes quite definite if we set some elements of the constituent vectors.

2. Examples of decomposition.

Consider a few very common cases of decomposition.

1. Decompose the given vector c into two component vectors of which one, for example a, is given in magnitude and direction.

The problem is reduced to determining the difference between two vectors. Indeed, if the vectors are components of the vector c, then the equality

From here, the second component vector is determined

2. Decompose the given vector c into two components, one of which must lie in a given plane and the other must lie on a given straight line a.

To determine the constituent vectors, we transfer the vector c so that its beginning coincides with the point of intersection of the given straight line with the plane (point O - see Fig. 18). From the end of the vector c (point C), draw a straight line to

intersection with the plane (B is the intersection point), and then from point C we draw a straight line parallel

The vectors and will be the sought ones, that is, it is natural that the indicated decomposition is possible if the straight line a and the plane are not parallel.

3. Three coplanar vectors a, b and c are given, and the vectors are not collinear. It is required to decompose the vector c into vectors

Let us bring all three given vectors to one point O. Then, due to their coplanarity, they will be located in one plane. On the given vector c, as on the diagonal, we construct a parallelogram, the sides of which are parallel to the lines of action of the vectors (Fig. 19). This construction is always possible (unless the vectors are collinear) and is unique. Fig. 19 shows that