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198 ALGEBRA

Properties of the set of all solutions of a homogeneous system:
1. All solutions of a homogeneous system (5.5.1.3) form a linear space.
2. The linear space of all solutions of a homogeneous system (5.5.1.3) with n unknown
quantities and a basic matrix of rank r is isomorphic to the space A n–r of all ordered
systems of (n – r) numbers. The dimension of the space of solutions is equal to n – r.
3. Anysystem of (n–r) linearlyindependent solutions of the homogeneous system (5.5.1.3)
forms a basis in the space of all its solutions and is called a fundamental system of
solutions of that system. The fundamental system of solutions corresponding to the
basis i 1 =(1, 0, ... , 0), i 2 =(0, 1, ... , 0), ... , i n–r =(0, 0, ... , 1)ofthe space A n–r is
said to be normal.

5.5.1-3. Consistency condition for a general linear system.
System (5.5.1.1) or (5.5.1.2) is associated with two matrices: the basic matrix A of size
m×n and the augmented matrix A 1 of size m×(n+1) formed by the matrix A supplemented
with the column of the free terms, i.e.,

⎛ ⎞
a 11 a 12 ··· a 1n b 1
⎜ a 21 a 22 ··· a 2n b 2 ⎟
A 1 ≡ ⎜ . . . . . ⎟ . (5.5.1.4)
⎝ . . . .
. . . . . . ⎠
a m1 a m2 ··· a mn b m
KRONECKER–CAPELLI THEOREM. A linear system (5.5.1.1) [or (5.5.1.2)] is consistent
if and only if its basic matrix and its augmented matrix (5.5.1.4) have the same rank, i.e.
rank (A 1 )= rank (A).

5.5.2. Finding Solutions of a System of Linear Equations

5.5.2-1. System of two equations with two unknown quantities.

A system of two equations with two unknown quantities has the form
a 1 x + b 1 y = c 1 ,
(5.5.2.1)
a 2 x + b 2 y = c 2 .

Depending on the coefficients a k , b k , c k , the following three cases are possible:

1 .If Δ = a 1 b 2 – a 2 b 1 ≠ 0, then system (5.5.2.1) has a unique solution,
◦
c 1 b 2 – c 2 b 1 a 1 c 2 – a 2 c 1
x = , y = .
a 1 b 2 – a 2 b 1 a 1 b 2 – a 2 b 1
2 .If Δ = a 1 b 2 – a 2 b 1 = 0 and a 1 c 2 – a 2 c 1 = 0 (the case of proportional coefficients), then
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system (5.5.2.1) has infinitely many solutions described by the formulas
c 1 – a 1 t
x = t, y = (b 1 ≠ 0),
b 1

where t is arbitrary.
3 .If Δ = a 1 b 2 – a 2 b 1 = 0 and a 1 c 2 – a 2 c 1 ≠ 0, then system (5.5.2.1) has no solutions.
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