28. Solubility of non-homogeneous systems of equations

According to Section 26, you must determine in the case of a non- homogeneous system of equations whether it is soluble and, if so, find anyone solution.

Beside the coefficient matrix a of the system of Equation (26.1), we will still consider the matrix A₁ which arises from A by addition of the column of the right hand sides

Denote by r and r₁ the Ranks of these matrices. Naturally, r r₁, since addition of a column can only increase the Rank.

The solubility of the system (26.1) is the subject of

Theorem 15

The system of equations (26.1) is soluble if and only if r = r₁, that is, if the Rank of the coefficient matrix A does not increase on addition of the column of the right hand sides.

Proof: Let to start with the system (26.1) be soluble and show that then r = r₁. Denote the columns of A by a₁, ··· , a_n. Then (26.1) can be rewritten

a₁x₁ + ··· + a_nx_n= c. (28.1)

The (p,1)-matrices a₁, ··· , a_n span in the vector space of all (p,1)-matrices a partial space A, the dimension of which equals the maximum number of linearly independent vectors a₁, ··· , a_n, that is, the Rank r of A. Correspondingly, the vectors a₁, ··· , a_n, c span a partial space A₁ of dimension r₁. Equation (28.1) states that c already belongs to the partial space A, because c can be expressed as a linear combination of the a₁, ··· , a_n. Consequently, A is not enlarged by addition of c, A and A₁ and, in particular, r = r₁. *

*Cf. the corresponding argument at the end of Section 23.

Conversely, let now r = r₁. We have to show that then (28.1)with suitable numbers x₁, ··· , x_n can be satisfied or, in other words, that c depends linearly on a₁, ··· , a_n. Because A is always a (possibly not genuine) partial space of A₁ and there does not occur an increase in dimension, due to the assumption r=r₁ during the transition from A to A₁, the spaces A and A₁ agree. Consequently, every vector of A₁ is also contained in A, that is, it can be represented as a linear sum of a₁, ··· , a_n. In particular, this is true for the vector c, whence the proof is complete.

If the condition r = r₁ is met, it is not difficult to find a solution of (26.1). The matrix A₁ contains r linearly independent rows. Again, no restriction of generality arises, if it is assumed that the first r rows of A₁ are linearly independent. All other rows of A₁ are then linear combinations of the first rows or, in other words, all Equations (26.1) follow already from the first r equations. Hence it is sufficient to solve the r equations

a₁₁x₁ + ··· + a_1nx_n = c₁,
····················································(28.2)
a_r1x₁ + ··· + a_rnx_n = c_r.

Since Rank A = r, there exists in A a minor, non-zero determinant of order r. As before, we introduce for the sake of simplicity the notation

We can now rewrite Equations (28.3):

a₁₁x₁ + ··· + a_1rx_r = c₁ - a_1,r+1x_r+1 - ··· - a_1nx_n,
·······················································································
a_r1x₁ + ··· + a_rrx_r = c_r - a_r,r+1x_r+1 - ··· - a_rnx_n.

If you substitute for the unknowns x_r+1, ··· , x_n arbitrary values, you can, due to (28.3), find the unknowns x₁, ··· , x_r by Cramer's Rule, so that the system of equations is satisfied. Hence we have found a solution of (26.1) with arbitrarily given x_r+1, ··· , x_n.

If not (28.3), but another minor determinant of order r of the coefficient matrix of (28.2) differ from zero, you have to shift other suitable unknowns to the right hand side. You only have to watch out that there occurs always on the left hand side a coefficient matrix with a non-zero determinant.

We will now find for Theorem 15 another formulation and indeed one in which there no longer appears the concept of Rank. In the new formulation, you can literally transfer the statement to integral equations or systems with infinitely many unknowns.

Form with a (p,1)-matrix y of p unknowns y₁, ··· , y_n the linear equations

These equations state that the vector y is orthogonal to the vectors a₁,···,a_n, c. The set of the vectors , which satisfy Equations (28.4), forms a partial space R₁ of the vector space of all (p,1)-matrices. Those vectors which satisfy only Equations (28.4) with exception of the last, form a vector space R, including R₁. By Theorem 14,

Dimension R₁ = p - r₁,
Dimension R = p - r.

As a consequence, r = r₁, which is equivalent to the coincidence of R and R₁, while in the case r < r₁ the space R is wider than R₁. Hence, in the first case, every vector orthogonal to a₁, ··· , a_n is also orthogonal to c, while for r < r₁ there exist vectors, which while orthogonal to a₁, ··· , a_n are not so to c. Hence, taking Theorem 15 into consideration, we can formulate the condition of solubility of a non-homogeneous system of equations as follows:

Theorem 16

The system of equations (26.1) has then and only then a solution, if every vector, which is orthogonal to all columns of the coefficient matrix, is also orthogonal to the column of the right hand sides.

A vector, which is orthogonal to the columns of the coefficient matrix in (26.1) is equivalent to a linear relation between the left hand sides of Equation (26.1). Theorem 16 states that the system of equations has a solution if and only if every linear relation between the left hand sides is also valid for the column of the right hand sides. This is a very informative result.

Also the solutions of a non-homogeneous system of equations is obtained solely by application of the four basic rules of calculation. Hence a non-homogeneous system of equations, which is on the whole soluble, has always solutions out of the smallest body of numbers, which contains the coefficients and right hand sides.

Beside a system of n homogeneous equations with n unknowns, we consider all non-homogeneous systems with the same left hand sides. Since the determinant of the coefficient matrix either does or does not vanish, there exists with regard to the solubility of homogeneous and non-homogeneous systems the alternative:

Either the homogeneous system has non-trivial solutions, when a non- homogeneous system is only soluble under additional conditions or the homogeneous system has only a trivial solution, when every non-homogenous system can be solved.

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