Geometric criterion for the linear dependence of three vectors. Necessary condition for linear dependence of n functions. Properties of linearly dependent vectors

Note that in what follows, without loss of generality, we will consider the case of vectors in three-dimensional space. On the plane, the consideration of vectors is carried out in a similar way. As noted above, all the results known from the course of linear algebra for algebraic vectors can be transferred to the particular case of geometric vectors. So let's do it.

Let vectors be fixed.

Definition. The sum, where are some numbers, is called a linear combination of vectors. In this case, these numbers will be called the coefficients of the linear combination.

We will be interested in the question of the possibility of equality of a linear combination to a zero vector. In accordance with the properties and axioms of vector spaces, it becomes obvious that for any system of vectors there is a trivial (zero) set of coefficients for which this equality holds:

The question arises of the existence for a given system of vectors of a non-trivial set of coefficients (among which there is at least one non-zero coefficient), for which the mentioned equality holds. In accordance with this, we will distinguish between linearly dependent and independent systems.

Definition. A system of vectors is called linearly independent if there is such a set of numbers , among which there is at least one non-zero one, such that the corresponding linear combination is equal to the zero vector:

A system of vectors is called linearly independent if the equality

is possible only in the case of a trivial set of coefficients:

Let us list the main properties of linearly dependent and independent systems proved in the course of linear algebra.

1. Any system of vectors containing a zero vector is linearly dependent.

2. Let there be a linearly dependent subsystem in the system of vectors. Then the whole system is also linearly dependent.

3. If a system of vectors is linearly independent, then any of its subsystems is also linearly independent.

4. If there are two vectors in a system of vectors, one of which is obtained from the other by multiplying by a certain number, then the entire system is linearly dependent.

Theorem (criterion of linear dependence). A system of vectors is linearly dependent if and only if one of the vectors of this system can be represented as a linear combination of the other vectors of the system.

Taking into account the criterion of collinearity of two vectors, it can be argued that the criterion for their linear dependence is their collinearity. For three vectors in space, the following statement is true.

Theorem (criterion for linear dependence of three geometric vectors). Three vectors , and are linearly dependent if and only if they are coplanar.

Proof.

Need. Let the vectors , and be linearly dependent. Let us prove their complanarity. Then, according to the general criterion of linear dependence of algebraic vectors, we assert that one of these vectors can be represented as a linear combination of other vectors. Let, for example,

If all three vectors , and are applied to a common origin , then the vector will coincide with the diagonal of the parallelogram built on the vectors and . But this means that the vectors , and lie in the same plane, i.e. coplanar.

Adequacy. Let the vectors , and be coplanar. Let us show that they are linearly dependent. First of all, consider the case when any pair of the indicated vectors is collinear. In this case, according to the previous theorem, the system of vectors , , contains a linearly dependent subsystem and, therefore, is itself linearly dependent according to property 2 of linearly dependent and independent systems of vectors. Let now no pair of vectors under consideration be collinear. We transfer all three vectors to one plane and bring them to a common origin. Draw through the end of the vector lines parallel to the vectors and . Let the letter denote the point of intersection of the line parallel to the vector with the line on which the vector lies, and by the letter the point of intersection of the line parallel to the vector with the line on which the vector lies. By definition of the sum of vectors, we get:

.

Since the vector is collinear to a nonzero vector , there exists a real number such that

Similar considerations imply the existence of a real number such that

As a result, we will have:

Then, from the general criterion for the linear dependence of algebraic vectors, we obtain that the vectors , , are linearly dependent. ■

Theorem (linear dependence of four vectors). Any four vectors are linearly dependent.

Proof. First of all, consider the case when any triple of the indicated four vectors is coplanar. In this case, this triple is linearly dependent in accordance with the previous theorem. Therefore, in accordance with the property of 2 linearly dependent and independent systems of vectors, and the entire quadruple is linearly dependent.

Let now, among the vectors under consideration, no triple of vectors be coplanar. Let us bring all four vectors , , , to a common beginning and draw planes through the end of the vector parallel to the planes defined by pairs of vectors , ; , ; , . The points of intersection of the indicated planes with the lines on which the vectors , and lie are denoted by the letters , and , respectively. It follows from the definition of the sum of vectors that

which, taking into account the general criterion of linear dependence of algebraic vectors, says that all four vectors are linearly dependent. ■

Def. System of elements x 1 ,…,x m lin. production V is called linearly dependent if ∃ λ 1 ,…, λ m ∈ ℝ (|λ 1 |+…+| λ m | ≠ 0) such that λ 1 x 1 +…+ λ m x m = θ .

Def. A system of elements x 1 ,…,x m ∈ V is called linearly independent if from the equality λ 1 x 1 +…+ λ m x m = θ ⟹λ 1 =…= λ m =0.

Def. An element x ∈ V is called a linear combination of elements x 1 ,…,x m ∈ V if ∃ λ 1 ,…, λ m ∈ ℝ such that x= λ 1 x 1 +…+ λ m x m .

Theorem (criterion of linear dependence): A system of vectors x 1 ,…,x m ∈ V is linearly dependent if and only if at least one vector of the system is linearly expressed in terms of the others.

Doc. Need: Let x 1 ,…,x m be linearly dependent ⟹ ∃ λ 1 ,…, λ m ∈ ℝ (|λ 1 |+…+| λ m | ≠ 0) such that λ 1 x 1 +…+ λ m -1 x m -1 + λ m x m = θ. Suppose λ m ≠ 0, then

x m \u003d (-) x 1 + ... + (-) x m -1.

Adequacy: Let at least one of the vectors be linearly expressed in terms of the other vectors: x m = λ 1 x 1 +…+ λ m -1 x m -1 (λ 1 ,…, λ m -1 ∈ ℝ) λ 1 x 1 +…+ λ m -1 x m -1 +(-1) x m =0 λ m =(-1) ≠ 0 ⟹ x 1 ,…,x m - are linearly independent.

Ven. linear dependence condition:

If the system contains a zero element or a linearly dependent subsystem, then it is linearly dependent.

λ 1 x 1 +…+ λ m x m = 0 – linearly dependent system

1) Let x 1 = θ, then this equality is valid for λ 1 =1 and λ 1 =…= λ m =0.

2) Let λ 1 x 1 +…+ λ m x m =0 be a linearly dependent subsystem ⟹|λ 1 |+…+| λ m | ≠ 0 . Then for λ 1 =0 we also obtain |λ 1 |+…+| λ m | ≠ 0 ⟹ λ 1 x 1 +…+ λ m x m =0 is a linearly dependent system.

Basis of a linear space. Vector coordinates in the given basis. The coordinates of the sums of vectors and the product of a vector by a number. Necessary and sufficient condition for linear dependence of a system of vectors.

Definition: An ordered system of elements e 1, ..., e n of a linear space V is called a basis of this space if:

A) e 1 ... e n are linearly independent

B) ∀ x ∈ α 1 … α n such that x= α 1 e 1 +…+ α n e n

x= α 1 e 1 +…+ α n e n – expansion of the element x in the basis e 1, …, e n

α 1 … α n ∈ ℝ are the coordinates of the element x in the basis e 1, …, e n

Theorem: If the basis e 1, …, e n is given in the linear space V, then ∀ x ∈ V the column of coordinates x in the basis e 1, …, e n is uniquely determined (the coordinates are uniquely determined)

Proof: Let x=α 1 e 1 +…+ α n e n and x=β 1 e 1 +…+β n e n

x= ⇔ = Θ, i.e. e 1, …, e n are linearly independent, then - =0 ∀ i=1, …, n ⇔ = ∀ i=1, …, n h.t.d.

Theorem: let e 1, …, e n be the basis of the linear space V; x, y are arbitrary elements of the space V, λ ∈ ℝ is an arbitrary number. When x and y are added, their coordinates are added, when x is multiplied by λ, the coordinates of x are also multiplied by λ.

Proof: x= (e 1, …, e n) and y= (e 1, …, e n)

x+y= + = (e 1, …, e n)

λx= λ ) = (e 1, …, e n)

Lemma1: (necessary and sufficient condition for the linear dependence of a system of vectors)

Let e ​​1 …e n be the basis of the space V. The system of elements f 1 , …, f k ∈ V is linearly dependent if and only if the coordinate columns of these elements in the basis e 1, …, e n are linearly dependent

Proof: expand f 1 , …, f k in the basis e 1, …, e n

f m =(e 1, …, e n) m=1, …, k

λ 1 f 1 +…+λ k f k =(e 1, …, e n)[ λ 1 +…+ λ n ] i.e. λ 1 f 1 +…+λ k f k = Θ ⇔

⇔ λ 1 +…+ λ n = as required.

13. Dimension of a linear space. Theorem on the relationship between dimension and basis.
Definition: A linear space V is called an n-dimensional space if there are n linearly independent elements in V, and a system of any n + 1 elements of the space V is linearly dependent. In this case, n is called the dimension of the linear space V and is denoted dimV=n.

A linear space is called infinite-dimensional if ∀N ∈ ℕ in the space V there exists a linearly independent system containing N elements.

Theorem: 1) If V is an n-dimensional linear space, then any ordered system of n linearly independent elements of this space forms a basis. 2) If in the linear space V there is a basis consisting of n elements, then the dimension of V is equal to n (dimV=n).

Proof: 1) Let dimV=n ⇒ in V ∃ n linearly independent elements e 1, …,e n . We prove that these elements form a basis, that is, we prove that ∀ x ∈ V can be expanded in terms of e 1, …,e n . Let's add x to them: e 1, …,e n , x – this system contains n+1 vectors, which means it is linearly dependent. Since e 1, …,e n is linearly independent, then by Theorem 2 x linearly expressed through e 1, …,e n i.e. ∃ ,…, such that x= α 1 e 1 +…+ α n e n . So e 1, …,e n is the basis of the space V. 2)Let e ​​1, …,e n be the basis of V, so there are n linearly independent elements in V ∃ n. Take arbitrary f 1 ,…,f n ,f n +1 ∈ V – n+1 elements. Let's show their linear dependence. Let's break them down in terms of:

f m =(e 1, …,e n) = where m = 1,…,n Let's create a matrix of coordinate columns: A= Matrix contains n rows ⇒ RgA≤n. Number of columns n+1 > n ≥ RgA ⇒ Columns of matrix A (ie columns of coordinates f 1 ,…,f n ,f n +1) are linearly dependent. From Lemma 1 ⇒ ,…,f n ,f n +1 are linearly dependent ⇒ dimV=n.

Consequence: If any basis contains n elements, then any other basis of this space contains n elements.

Theorem 2: If the system of vectors x 1 ,… ,x m -1 , x m is linearly dependent, and its subsystem x 1 ,… ,x m -1 is linearly independent, then x m - is linearly expressed through x 1 ,… ,x m -1

Proof: Because x 1 ,… ,x m -1 , x m is linearly dependent, then ∃ , …, , ,

, …, | , | such that . If , , …, | => x 1 ,… ,x m -1 are linearly independent, which cannot be. So m = (-) x 1 +…+ (-) x m -1.

The following give several criteria for linear dependence and, accordingly, linear independence of systems of vectors.

Theorem. (A necessary and sufficient condition for the linear dependence of vectors.)

A system of vectors is dependent if and only if one of the vectors of the system is linearly expressed in terms of the others of this system.

Proof. Need. Let the system be linearly dependent. Then, by definition, it represents the null vector in a non-trivial way, i.e. there is a non-trivial combination of this system of vectors equal to the zero vector:

where at least one of the coefficients of this linear combination is not equal to zero. Let , .

Divide both parts of the previous equality by this non-zero coefficient (i.e. multiply by:

Denote: , where .

those. one of the vectors of the system is linearly expressed in terms of the others of this system, etc.

Adequacy. Let one of the vectors of the system be linearly expressed in terms of other vectors of this system:

Let's move the vector to the right of this equality:

Since the coefficient of the vector is , then we have a non-trivial representation of zero by the system of vectors , which means that this system of vectors is linearly dependent, etc.

The theorem has been proven.

Consequence.

1. A system of vectors in a vector space is linearly independent if and only if none of the vectors of the system is linearly expressed in terms of other vectors of this system.

2. A system of vectors containing a zero vector or two equal vectors is linearly dependent.

Proof.

1) Necessity. Let the system be linearly independent. Assume the opposite and there is a system vector that is linearly expressed through other vectors of this system. Then, by the theorem, the system is linearly dependent, and we arrive at a contradiction.

Adequacy. Let none of the vectors of the system be expressed in terms of others. Let's assume the opposite. Let the system be linearly dependent, but then it follows from the theorem that there is a system vector that is linearly expressed through other vectors of this system, and we again come to a contradiction.

2a) Let the system contain a zero vector. Assume for definiteness that the vector :. Then the equality

those. one of the vectors of the system is linearly expressed in terms of the other vectors of this system. It follows from the theorem that such a system of vectors is linearly dependent, so on.

Note that this fact can be proved directly from a linearly dependent system of vectors.

Since , the following equality is obvious

This is a non-trivial representation of the zero vector, which means that the system is linearly dependent.

2b) Let the system have two equal vectors. Let for . Then the equality

Those. the first vector is linearly expressed in terms of the other vectors of the same system. It follows from the theorem that the given system is linearly dependent, and so on.

Similarly to the previous one, this assertion can also be proved directly from the definition of a linearly dependent system.

A necessary and sufficient condition for the linear dependence of two

vectors is their collinearity.

2. Scalar product- an operation on two vectors, the result of which is a scalar (number) that does not depend on the coordinate system and characterizes the lengths of the multiplier vectors and the angle between them. This operation corresponds to the multiplication length given vector x on projection another vector y to the given vector x. This operation is usually viewed as commutative and linear in each factor.

Dot product properties:

3. Three vectors (or more) are called coplanar if they, being reduced to a common origin, lie in the same plane.

A necessary and sufficient condition for the linear dependence of three vectors is their coplanarity. Any four vectors are linearly dependent. basis in space any ordered triple of non-coplanar vectors is called. A basis in space allows one to unambiguously associate with each vector an ordered triple of numbers - the coefficients of the representation of this vector in a linear combination of vectors of the basis. On the contrary, with the help of a basis, we will associate a vector with each ordered triplet of numbers if we make a linear combination. An orthogonal basis is called orthonormal , if its vectors are equal to one in length. For an orthonormal basis in space, the notation is often used. Theorem: In an orthonormal basis, the coordinates of vectors are the corresponding orthogonal projections of this vector onto the directions of the coordinate vectors. A triple of non-coplanar vectors a, b, c called right, if the observer from their common origin bypasses the ends of the vectors a, b, c in that order seems to proceed clockwise. Otherwise a, b, c - left triple. All right (or left) triples of vectors are called equally oriented. A rectangular coordinate system on a plane is formed by two mutually perpendicular coordinate axes OX and OY. The coordinate axes intersect at a point O, which is called the origin, each axis has a positive direction. AT right hand coordinate system, the positive direction of the axes is chosen so that with the direction of the axis OY up, axis OX looked to the right.

Four angles (I, II, III, IV) formed by the coordinate axes X"X and Y"Y, are called coordinate angles or quadrants(see fig. 1).

if vectors and with respect to an orthonormal basis on the plane have coordinates and, respectively, then the scalar product of these vectors is calculated by the formula

4. Vector product of two vectors a and b is an operation on them, defined only in three-dimensional space, the result of which is vector with the following

properties:

The geometric meaning of the cross product of vectors is the area of ​​a parallelogram built on vectors. A necessary and sufficient condition for the collinarity of a nonzero vector and a vector is the existence of a number that satisfies the equality .

If two vectors and are defined by their rectangular Cartesian coordinates, or more precisely, they are represented in a vorthonormalized basis

and the coordinate system is right, then their vector product has the form

To remember this formula, it is convenient to use the determinant:

5. Mixed product vectors - the scalar product of a vector and the cross product of vectors and :

Sometimes it is called triple scalar product vectors, apparently due to the fact that the result is a scalar (more precisely, a pseudoscalar).

Geometric sense: The module of the mixed product is numerically equal to the volume of the parallelepiped formed by the vectors .

When two factors are interchanged, the mixed product changes sign to the opposite:

With a cyclic (circular) permutation of factors, the mixed product does not change:

The mixed product is linear in any factor.

The mixed product is zero if and only if the vectors are coplanar.

1. Complanarity condition for vectors: three vectors are coplanar if and only if their mixed product is zero.

§ A triple of vectors containing a pair of collinear vectors is coplanar.

§ Mixed product of coplanar vectors. This is a criterion for the coplanarity of three vectors.

§ Coplanar vectors are linearly dependent. This is also a criterion for coplanarity.

§ There are real numbers such that for coplanar , except for or . This is a reformulation of the previous property and is also a criterion for coplanarity.

§ In a 3-dimensional space, 3 non-coplanar vectors form a basis. That is, any vector can be represented as: . Then will be the coordinates in the given basis.

The mixed product in the right Cartesian coordinate system (in the orthonormal basis) is equal to the determinant of the matrix composed of the vectors and :

§6. General equation (complete) of the plane

where and are constants, moreover, and are not equal to zero at the same time; in vector form:

where is the radius vector of the point , the vector is perpendicular to the plane (normal vector). Direction cosines vector :

If one of the coefficients in the plane equation is zero, the equation is called incomplete. When the plane passes through the origin of coordinates, when (or , ) P. is parallel to the axis (respectively or ). For ( , or ), the plane is parallel to the plane (or , respectively).

§ Equation of a plane in segments:

where , , are the segments cut off by the plane on the axes and .

§ Equation of a plane passing through a point perpendicular to the normal vector :

in vector form:

(mixed product of vectors), otherwise

§ Normal (normalized) plane equation

§ Angle between two planes. If the P. equations are given in the form (1), then

If in vector form, then

§ Planes are parallel, if

Or (Vector Product)

§ Planes are perpendicular, if

Or . (Scalar product)

7. Equation of a plane passing through three given points , not lying on the same line:

8. The distance from a point to a plane is the smallest of the distances between this point and the points of the plane. It is known that the distance from a point to a plane is equal to the length of the perpendicular dropped from this point to the plane.

§ Point Deviation from the plane given by the normalized equation

If and the origin lie on opposite sides of the plane, otherwise . The distance from a point to a plane is

§ The distance from the point to the plane given by the equation is calculated by the formula:

9. Plane bundle- the equation of any P. passing through the line of intersection of two planes

where α and β are any numbers not simultaneously equal to zero.

In order for the three planes defined by their general equations A 1 x+B 1 y+C 1 z+D 1 =0, A 2 x+B 2 y+C 2 z+D 2 =0, A 3 x+B 3 y+C 3 z+D 3 =0 with respect to PDSC belonged to one beam, proper or improper, it is necessary and sufficient that the rank of the matrix be equal to either two or one.
Theorem 2. Let two planes π 1 and π 2 be given with respect to PDSC by their general equations: A 1 x+B 1 y+C 1 z+D 1 =0, A 2 x+B 2 y+C 2 z+D 2 = 0. In order for the π 3 plane, given relative to the PDSC by its general equation A 3 x+B 3 y+C 3 z+D 3 =0, to belong to the beam formed by the π 1 and π 2 planes, it is necessary and sufficient that the left side of the equation of the plane π 3 was represented as a linear combination of the left parts of the equations of the planes π 1 and π 2 .

10.Vector parametric equation of a straight line in space:

where is the radius vector of some fixed point M 0 lying on a straight line is a non-zero vector collinear to this straight line, is the radius vector of an arbitrary point on the straight line.

Parametric equation of a straight line in space:

M

Canonical equation of a straight line in space:

where are the coordinates of some fixed point M 0 lying on a straight line; - coordinates of a vector collinear to this line.

General vector equation of a straight line in space:

Since the line is the intersection of two different non-parallel planes, given respectively by the general equations:

then the equation of a straight line can be given by a system of these equations:

The angle between the direction vectors and will be equal to the angle between the lines. The angle between vectors is found using the scalar product. cosA=(ab)/IaI*IbI

The angle between a straight line and a plane is found by the formula:

where (A; B; C;) are the coordinates of the normal vector of the plane
(l;m;n;) directing vector coordinates of the straight line

Conditions for parallelism of two lines:

a) If the lines are given by equations (4) with a slope, then the necessary and sufficient condition for their parallelism is the equality of their slopes:

k 1 = k 2 . (8)

b) For the case when the lines are given by equations in general form (6), the necessary and sufficient condition for their parallelism is that the coefficients at the corresponding current coordinates in their equations are proportional, i.e.

Conditions for perpendicularity of two lines:

a) In the case when the lines are given by equations (4) with a slope, the necessary and sufficient condition for their perpendicularity is that their slopes are reciprocal in magnitude and opposite in sign, i.e.

b) If the equations of straight lines are given in general form (6), then the condition for their perpendicularity (necessary and sufficient) is to fulfill the equality

A 1 A 2 + B 1 B 2 = 0. (12)

A line is said to be perpendicular to a plane if it is perpendicular to any line in that plane. If a line is perpendicular to each of two intersecting lines of a plane, then it is perpendicular to that plane. In order for a line and a plane to be parallel, it is necessary and sufficient that the normal vector to the plane and the directing vector of the line be perpendicular. For this, it is necessary that their scalar product be equal to zero.

For a line and a plane to be perpendicular, it is necessary and sufficient that the normal vector to the plane and the directing vector of the line be collinear. This condition is satisfied if the cross product of these vectors was equal to zero.

12. In space, the distance from a point to a straight line given by a parametric equation

can be found as the minimum distance from a given point to an arbitrary point on a straight line. Coefficient t this point can be found by the formula

Distance between intersecting lines is the length of their common perpendicular. It is equal to the distance between parallel planes passing through these lines.

Necessary condition for linear dependence of n functions.

Let the functions , have derivatives of the limit (n-1).

Consider the determinant: (1)

W(x) is usually called the Wronsky determinant for functions .

Theorem 1. If the functions are linearly dependent in the interval (a,b), then their Wronskian W(x) is identically equal to zero in this interval.

Proof. By the condition of the theorem, the relation

, (2) where not all are equal to zero. Let . Then

(3). Differentiate this identity n-1 times and,

substituting instead of their obtained values ​​into the Vronsky determinant,

we get:

In the Wronsky determinant, the last column is a linear combination of the previous n-1 columns and therefore equals zero at all points of the interval (a,b).

Theorem 2. If the functions y 1 ,..., y n are linearly independent solutions of the equation L[y] = 0, all coefficients of which are continuous in the interval (a,b), then the Wronskian of these solutions is different from zero at every point interval (a,b).

Proof. Let's assume the opposite. There is X 0 , where W(X 0)=0. We compose a system of n equations

Obviously, system (5) has a nonzero solution. Let (6).

Let us compose a linear combination of solutions y 1 ,..., y n .

Y(x) is a solution to the equation L[y] = 0. In addition, . By virtue of the uniqueness theorem, the solution of the equation L[y] = 0 with zero initial conditions must be only zero, ᴛ.ᴇ. .

We get the identity , where not all are equal to zero, which means that y 1 ,..., y n are linearly dependent, which contradicts the condition of the theorem. Therefore, there is no such point where W(X 0)=0.

Based on Theorem 1 and Theorem 2, we can formulate the following assertion. For n solutions of the equation L[y] = 0 to be linearly independent in the interval (a,b), it is extremely important and sufficient that their Wronskian does not vanish at any point of this interval.

The following obvious properties of the Wronskian also follow from the proved theorems.

1. If the Wronskian of n solutions of the equation L[y] = 0 is equal to zero at one point x = x 0 from the interval (a,b), in which all the coefficients p i (x) are continuous, then it is equal to zero at all ex points of this interval.
2. If the Wronskian of n solutions of the equation L[y] = 0 is nonzero at one point x = x 0 from the interval (a,b), then it is nonzero at all points of this interval.

Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, for the linearity of n independent solutions of the equation L[y] = 0 in the interval (a,b), in which the coefficients of the equation p i (x) are continuous, it is extremely important and sufficient that their Wronskian be different from zero even in one point of this interval.

Necessary condition for linear dependence of n functions. - concept and types. Classification and features of the category "A necessary condition for the linear dependence of n functions." 2017, 2018.

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