Solution find using a calculator. The solution algorithm is the same as for systems of linear not homogeneous equations.
Operating only with rows, we find the rank of the matrix, the basis minor; We declare dependent and free unknowns and find a general solution.
The first and second lines are proportional, let’s cross out one of them:
.
Dependent variables – x 2, x 3, x 5, free – x 1, x 4. From the first equation 10x 5 = 0 we find x 5 = 0, then
; .
The general solution is:
We find a fundamental system of solutions, which consists of (n-r) solutions. In our case, n=5, r=3, therefore, the fundamental system of solutions consists of two solutions, and these solutions must be linearly independent. For the rows to be linearly independent, it is necessary and sufficient that the rank of the matrix composed of the elements of the rows be equal to the number of rows, that is, 2. It is enough to give the free unknowns x 1 and x 4 values from the rows of the second-order determinant, nonzero, and calculate x 2 , x 3 , x 5 . The simplest non-zero determinant is .
So the first solution is:
, second – 
.
These two decisions constitute a fundamental decision system. Note that the fundamental system is not unique (you can create as many nonzero determinants as you like).
Example 2. Find the general solution and fundamental system of solutions of the system 
Solution.
,
it follows that the rank of the matrix is 3 and equal to the number of unknowns. This means that the system does not have free unknowns, and therefore has a unique solution - a trivial one.
Exercise . Explore and solve the system linear equations.
Example 4
Exercise . Find the general and particular solutions of each system.
Solution. Let's write down the main matrix of the system:
| 5 | -2 | 9 | -4 | -1 |
| 1 | 4 | 2 | 2 | -5 |
| 6 | 2 | 11 | -2 | -6 |
| x 1 | x 2 | x 3 | x 4 | x 5 |
Let's reduce the matrix to triangular form. We will work only with rows, since multiplying a matrix row by a number other than zero and adding it to another row for the system means multiplying the equation by the same number and adding it with another equation, which does not change the solution of the system.
Multiply the 2nd line by (-5). Let's add the 2nd line to the 1st:
| 0 | -22 | -1 | -14 | 24 |
| 1 | 4 | 2 | 2 | -5 |
| 6 | 2 | 11 | -2 | -6 |
Let's multiply the 2nd line by (6). Multiply the 3rd line by (-1). Let's add the 3rd line to the 2nd:
Let's find the rank of the matrix.
| 0 | 22 | 1 | 14 | -24 |
| 6 | 2 | 11 | -2 | -6 |
| x 1 | x 2 | x 3 | x 4 | x 5 |
The selected minor has the highest order (of possible minors) and is non-zero (it is equal to the product of the elements on the reverse diagonal), therefore rang(A) = 2.
This minor is basic. It includes coefficients for the unknowns x 1 , x 2 , which means that the unknowns x 1 , x 2 are dependent (basic), and x 3 , x 4 , x 5 are free.
Let's transform the matrix, leaving only the basis minor on the left.
| 0 | 22 | 14 | -1 | -24 |
| 6 | 2 | -2 | -11 | -6 |
| x 1 | x 2 | x 4 | x 3 | x 5 |
The system with the coefficients of this matrix is equivalent to the original system and has the form:
22x 2 = 14x 4 - x 3 - 24x 5
6x 1 + 2x 2 = - 2x 4 - 11x 3 - 6x 5
Using the method of eliminating unknowns, we find non-trivial solution:
We obtained relations expressing the dependent variables x 1 , x 2 through the free ones x 3 , x 4 , x 5 , that is, we found common decision:
x 2 = 0.64x 4 - 0.0455x 3 - 1.09x 5
x 1 = - 0.55x 4 - 1.82x 3 - 0.64x 5
We find a fundamental system of solutions, which consists of (n-r) solutions.
In our case, n=5, r=2, therefore, the fundamental system of solutions consists of 3 solutions, and these solutions must be linearly independent.
For the rows to be linearly independent, it is necessary and sufficient that the rank of the matrix composed of row elements be equal to the number of rows, that is, 3.
It is enough to give the free unknowns x 3 , x 4 , x 5 values from the lines of the 3rd order determinant, non-zero, and calculate x 1 , x 2 .
The simplest non-zero determinant is the identity matrix.
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
Task . Find a fundamental set of solutions to a homogeneous system of linear equations.
Homogeneous system of linear equations over a field
DEFINITION. A fundamental system of solutions to a system of equations (1) is a non-empty linearly independent system of its solutions, the linear span of which coincides with the set of all solutions to system (1).
Note that a homogeneous system of linear equations that has only a zero solution does not have fundamental system decisions.
PROPOSAL 3.11. Any two fundamental systems of solutions to a homogeneous system of linear equations consist of the same number decisions.
Proof. In fact, any two fundamental systems of solutions to the homogeneous system of equations (1) are equivalent and linearly independent. Therefore, by Proposition 1.12, their ranks are equal. Consequently, the number of solutions included in one fundamental system is equal to the number of solutions included in any other fundamental system of solutions.
If the main matrix A of the homogeneous system of equations (1) is zero, then any vector from is a solution to system (1); in this case, any set of linearly independent vectors from is a fundamental system of solutions. If the column rank of matrix A is equal to , then system (1) has only one solution - zero; therefore, in this case, the system of equations (1) does not have a fundamental system of solutions.
THEOREM 3.12. If the rank of the main matrix of a homogeneous system of linear equations (1) is less than the number of variables , then system (1) has a fundamental solution system consisting of solutions.
Proof. If the rank of the main matrix A of the homogeneous system (1) is equal to zero or , then it was shown above that the theorem is true. Therefore, below it is assumed that Assuming , we will assume that the first columns of matrix A are linearly independent. In this case, matrix A is rowwise equivalent to a reduced stepwise matrix, and system (1) is equivalent to the following reduced stepwise system of equations:
It is easy to check that any system of values of free variables of system (2) corresponds to one and only one solution to system (2) and, therefore, to system (1). In particular, only the zero solution of system (2) and system (1) corresponds to a system of zero values.
In system (2) we will assign one of the free variables a value equal to 1, and the remaining variables - zero values. As a result, we obtain solutions to the system of equations (2), which we write in the form of rows of the following matrix C:
The row system of this matrix is linearly independent. Indeed, for any scalars from the equality
equality follows
and, therefore, equality
Let us prove that the linear span of the system of rows of the matrix C coincides with the set of all solutions to system (1).
Arbitrary solution of system (1). Then the vector
is also a solution to system (1), and
The linear equation is called homogeneous, if its free term is equal to zero, and inhomogeneous otherwise. A system consisting of homogeneous equations is called homogeneous and has general form:
It is obvious that every homogeneous system is consistent and has a zero (trivial) solution. Therefore, when applied to homogeneous systems of linear equations, one often has to look for an answer to the question of the existence of nonzero solutions. The answer to this question can be formulated as the following theorem.
Theorem . A homogeneous system of linear equations has a nonzero solution if and only if its rank is less than the number of unknowns .
Proof: Let us assume that a system whose rank is equal has a non-zero solution. Obviously it does not exceed . In case the system has a unique solution. Since a system of homogeneous linear equations always has a zero solution, then the zero solution will be this unique solution. Thus, non-zero solutions are possible only for .
Corollary 1 : A homogeneous system of equations, in which the number of equations is less than the number of unknowns, always has a non-zero solution.
Proof: If a system of equations has , then the rank of the system does not exceed the number of equations, i.e. . Thus, the condition is satisfied and, therefore, the system has a non-zero solution.
Corollary 2 : A homogeneous system of equations with unknowns has a nonzero solution if and only if its determinant is zero.
Proof: Let us assume that a system of linear homogeneous equations, the matrix of which with the determinant , has a non-zero solution. Then, according to the proven theorem, and this means that the matrix is singular, i.e. .
Kronecker-Capelli theorem: An SLU is consistent if and only if the rank of the system matrix is equal to the rank of the extended matrix of this system. A system ur is called consistent if it has at least one solution.Homogeneous system of linear algebraic equations.
A system of m linear equations with n variables is called a system of linear homogeneous equations if all free terms are equal to 0. A system of linear homogeneous equations is always consistent, because it always has at least a zero solution. A system of linear homogeneous equations has a non-zero solution if and only if the rank of its matrix of coefficients for variables is less than the number of variables, i.e. for rank A (n. Any linear combination
Lin system solutions. homogeneous. ur-ii is also a solution to this system.
A system of linear independent solutions e1, e2,...,еk is called fundamental if each solution of the system is a linear combination of solutions. Theorem: if the rank r of the matrix of coefficients for the variables of a system of linear homogeneous equations is less than the number of variables n, then every fundamental system of solutions to the system consists of n-r solutions. Therefore, the general solution of the linear system. one-day ur-th has the form: c1e1+c2e2+...+skek, where e1, e2,..., ek is any fundamental system of solutions, c1, c2,...,ck are arbitrary numbers and k=n-r. The general solution of a system of m linear equations with n variables is equal to the sum
of the general solution of the system corresponding to it is homogeneous. linear equations and an arbitrary particular solution of this system.
7. Linear spaces. Subspaces. Basis, dimension. Linear shell. Linear space is called n-dimensional, if it contains a system of linearly independent vectors, and any system of a larger number of vectors is linearly dependent. The number is called dimension (number of dimensions) linear space and is denoted by . In other words, the dimension of space is maximum number linearly independent vectors of this space. If such a number exists, then the space is called finite-dimensional. If for anyone natural number n in space there is a system consisting of linearly independent vectors, then such a space is called infinite-dimensional (written: ). In what follows, unless otherwise stated, finite-dimensional spaces will be considered.
The basis of an n-dimensional linear space is an ordered collection of linearly independent vectors ( basis vectors).
Theorem 8.1 on the expansion of a vector in terms of a basis. If is the basis of an n-dimensional linear space, then any vector can be represented as a linear combination of basis vectors:
V=v1*e1+v2*e2+…+vn+en
and, moreover, in the only way, i.e. the coefficients are determined uniquely. In other words, any vector of space can be expanded into a basis and, moreover, in a unique way.
Indeed, the dimension of space is . The system of vectors is linearly independent (this is a basis). After adding any vector to the basis, we obtain linearly dependent system(since this system consists of vectors of n-dimensional space). Using the property of 7 linearly dependent and linearly independent vectors, we obtain the conclusion of the theorem.
Let M 0 – set of solutions to a homogeneous system (4) of linear equations.
Definition 6.12. Vectors With 1 ,With 2 , …, with p, which are solutions of a homogeneous system of linear equations are called fundamental set of solutions(abbreviated FNR), if
1) vectors With 1 ,With 2 , …, with p linearly independent (i.e., none of them can be expressed in terms of the others);
2) any other solution to a homogeneous system of linear equations can be expressed in terms of solutions With 1 ,With 2 , …, with p.
Note that if With 1 ,With 2 , …, with p– any f.n.r., then the expression k 1× With 1 + k 2× With 2 + … + k p× with p you can describe the whole set M 0 solutions to system (4), so it is called general view of the system solution (4).
Theorem 6.6. Any indeterminate homogeneous system of linear equations has a fundamental set of solutions.
The way to find the fundamental set of solutions is as follows:
Find a general solution to a homogeneous system of linear equations;
Build ( n – r) partial solutions of this system, while the values of the free unknowns must form identity matrix;
Write down the general form of the solution included in M 0 .
Example 6.5. Find a fundamental set of solutions to the following system:
Solution. Let's find a general solution to this system.
~ 
~ 
~ ~ 
Þ 
Þ Þ 
There are five unknowns in this system ( n= 5), of which there are two main unknowns ( r= 2), there are three free unknowns ( n – r), that is, the fundamental solution set contains three solution vectors. Let's build them. We have x 1 and x 3 – main unknowns, x 2 , x 4 , x 5 – free unknowns
Values of free unknowns x 2 , x 4 , x 5 form the identity matrix E third order. Got that vectors With 1 ,With 2 , With 3 form f.n.r. of this system. Then the set of solutions of this homogeneous system will be M 0 = {k 1× With 1 + k 2× With 2 + k 3× With 3 , k 1 , k 2 , k 3 О R).
Let us now find out the conditions for the existence of nonzero solutions of a homogeneous system of linear equations, in other words, the conditions for the existence of a fundamental set of solutions.
A homogeneous system of linear equations has non-zero solutions, that is, it is uncertain if
1) the rank of the main matrix of the system is less than the number of unknowns;
2) in a homogeneous system of linear equations, the number of equations is less than the number of unknowns;
3) if in a homogeneous system of linear equations the number of equations is equal to the number of unknowns, and the determinant of the main matrix is equal to zero (i.e. | A| = 0).
Example 6.6. At what parameter value a homogeneous system of linear equations 
has non-zero solutions?
Solution. Let's compose the main matrix of this system and find its determinant: = = 1×(–1) 1+1 × = – A– 4. The determinant of this matrix is equal to zero at a = –4.
Answer: –4.
7. Arithmetic n-dimensional vector space
Basic Concepts
In previous sections we have already encountered the concept of a set of real numbers located in in a certain order. This is a row matrix (or column matrix) and a solution to a system of linear equations with n unknown. This information can be summarized.
Definition 7.1. n-dimensional arithmetic vector called an ordered set of n real numbers.
Means A= (a 1 , a 2 , …, a n), where a iО R, i = 1, 2, …, n– general view of the vector. Number n called dimension vectors, and numbers a i are called his coordinates.
For example: A= (1, –8, 7, 4, ) – five-dimensional vector.
All set n-dimensional vectors are usually denoted as Rn.
Definition 7.2. Two vectors A= (a 1 , a 2 , …, a n) And b= (b 1 , b 2 , …, b n) of the same dimension equal if and only if their corresponding coordinates are equal, i.e. a 1 = b 1 , a 2 = b 2 , …, a n= b n.
Definition 7.3.Amount two n-dimensional vectors A= (a 1 , a 2 , …, a n) And b= (b 1 , b 2 , …, b n) is called a vector a + b= (a 1 + b 1, a 2 + b 2, …, a n+b n).
Definition 7.4. The work real number k to vector A= (a 1 , a 2 , …, a n) is called a vector k× A = (k×a 1, k×a 2 , …, k×a n)
Definition 7.5. Vector O= (0, 0, …, 0) is called zero(or null vector).
It is easy to verify that the actions (operations) of adding vectors and multiplying them by a real number have the following properties: " a, b, c Î Rn, " k, lО R:
1) a + b = b + a;
2) a + (b+ c) = (a + b) + c;
3) a + O = a;
4) a+ (–a) = O;
5) 1× a = a, 1 О R;
6) k×( l× a) = l×( k× a) = (l× k)× a;
7) (k + l)× a = k× a + l× a;
8) k×( a + b) = k× a + k× b.
Definition 7.6. A bunch of Rn with the operations of adding vectors and multiplying them by a real number given on it is called arithmetic n-dimensional vector space.
A system of linear equations in which all free terms are equal to zero is called homogeneous :
Any homogeneous system is always consistent, since it always has zero (trivial ) solution. The question arises under what conditions will a homogeneous system have a nontrivial solution.
Theorem 5.2.A homogeneous system has a nontrivial solution if and only if the rank of the underlying matrix is less than the number of its unknowns.
Consequence. A square homogeneous system has a nontrivial solution if and only if the determinant of the main matrix of the system is not equal to zero.
Example 5.6. Determine the values of the parameter l at which the system has nontrivial solutions, and find these solutions:
Solution. This system will have a non-trivial solution when the determinant of the main matrix is equal to zero:
Thus, the system is non-trivial when l=3 or l=2. For l=3, the rank of the main matrix of the system is 1. Then, leaving only one equation and assuming that y=a And z=b, we get x=b-a, i.e.
For l=2, the rank of the main matrix of the system is 2. Then, choosing the minor as the basis:
we get a simplified system
From here we find that x=z/4, y=z/2. Believing z=4a, we get
The set of all solutions of a homogeneous system has a very important linear property : if columns X 1 and X 2 - solutions to a homogeneous system AX = 0, then any linear combination of them a X 1 + b X 2 will also be a solution to this system. Indeed, since AX 1 = 0 And AX 2 = 0 , That A(a X 1 + b X 2) = a AX 1 + b AX 2 = a · 0 + b · 0 = 0. It is because of this property that if a linear system has more than one solution, then there will be an infinite number of these solutions.
Linearly independent columns E 1 , E 2 , Ek, which are solutions of a homogeneous system, are called fundamental system of solutions homogeneous system of linear equations if the general solution of this system can be written as a linear combination of these columns:
If a homogeneous system has n variables, and the rank of the main matrix of the system is equal to r, That k = n-r.
Example 5.7. Find the fundamental system of solutions to the following system of linear equations:
Solution. Let's find the rank of the main matrix of the system:
Thus, the set of solutions to this system of equations forms a linear subspace of dimension n-r= 5 - 2 = 3. Let’s choose minor as the base
.
Then, leaving only the basic equations (the rest will be a linear combination of these equations) and the basic variables (we move the rest, the so-called free variables to the right), we obtain a simplified system of equations:
Believing x 3 = a, x 4 = b, x 5 = c, we find
, 
.
Believing a= 1, b = c= 0, we obtain the first basic solution; believing b= 1, a = c= 0, we obtain the second basic solution; believing c= 1, a = b= 0, we obtain the third basic solution. As a result, the normal fundamental system of solutions will take the form
Using the fundamental system, the general solution of a homogeneous system can be written as
X = aE 1 + bE 2 + cE 3. a
Let us note some properties of solutions to an inhomogeneous system of linear equations AX=B and their relationship with the corresponding homogeneous system of equations AX = 0.
General solution of an inhomogeneous systemis equal to the sum of the general solution of the corresponding homogeneous system AX = 0 and an arbitrary particular solution of the inhomogeneous system. Indeed, let Y 0 is an arbitrary particular solution of an inhomogeneous system, i.e. AY 0 = B, And Y- general solution of a heterogeneous system, i.e. AY=B. Subtracting one equality from the other, we get
A(Y-Y 0) = 0, i.e. Y-Y 0 is the general solution of the corresponding homogeneous system AX=0. Hence, Y-Y 0 = X, or Y=Y 0 + X. Q.E.D.
Let the inhomogeneous system have the form AX = B 1 + B 2 . Then the general solution of such a system can be written as X = X 1 + X 2 , where AX 1 = B 1 and AX 2 = B 2. This property expresses the universal property of any linear systems(algebraic, differential, functional, etc.). In physics this property is called superposition principle, in electrical and radio engineering - principle of superposition. For example, in the theory of linear electrical circuits, the current in any circuit can be obtained as the algebraic sum of the currents caused by each energy source separately.