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5.5. SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS 197

Moreover, the vector x p is the unique element of V 1 for which the difference x – x p is
orthogonal to any element x 1 of V 1 , i.e.,

(x – x p ) ⋅ x 1 = 0 for all x 1 V 1 .

The mapping x → x p is a bounded linear operator from V to V 1 called the orthogonal
projection of the space V to its subspace V 1 .

5.5. Systems of Linear Algebraic Equations
5.5.1. Consistency Condition for a Linear System

5.5.1-1. Notion of a system of linear algebraic equations.
A system of m linear equations with n unknown quantities has the form
a 11 x 1 + a 12 x 2 + ··· + a 1k x k + ··· + a 1n x n = b 1 ,
a 21 x 1 + a 22 x 2 + ··· + a 2k x k + ··· + a 2n x n = b 2 ,
(5.5.1.1)
.. ........ ......... ........ ........ ........ ......
a m1 x 1 + a m2 x 2 + ··· + a mk x k + ··· + a mn x n = b m ,
where a 11 , a 12 , ... , a mn are the coefficients of the system; b 1 , b 2 , ... , b m are its free terms;
and x 1 , x 2 , ... , x n are the unknown quantities.
System (5.5.1.1) is said to be homogeneous if all its free terms are equal to zero. Other-
wise (i.e., if there is at least one nonzero free term) the system is called nonhomogeneous.
If the number of equations is equal to that of the unknown quantities (m = n), sys-
tem (5.5.1.1) is called a square system.
A solution of system (5.5.1.1) is a set of n numbers x 1 , x 2 , ... , x n satisfying the
equations of the system. A system is said to be consistent if it admits at least one solution.
If a system has no solutions, it is said to be inconsistent. A consistent system of the
form (5.5.1.1) is called a determined system—it has a unique solution. A consistent system
with more than one solution is said to be underdetermined.
It is convenient to use matrix notation for systems of the form (5.5.1.1),
AX = B, (5.5.1.2)

where A ≡ [a ij ] is a matrix of size m × n called the basic matrix of the system; X ≡ [x i ]is
a column vector of size n; B ≡ [b i ] is a column vector of size m.

5.5.1-2. Existence of nontrivial solutions of a homogeneous system.
Consider a homogeneous system
AX = O m , (5.5.1.3)
where A ≡ [a ij ] is its basic matrix of size m × n, X ≡ [x i ] is a column vector of size n,
and O m ≡ [0] is a column vector of size m. System (5.5.1.3) is always consistent since it
always has the so-called trivial solution X ≡ O n .
THEOREM. A homogeneous system (5.5.1.3) has a nontrivial solution if and only if the
rank of the matrix A is less than the number of the unknown quantities n.
It follows that a square homogeneous system has a nontrivial solution if and only if the
determinant of its matrix of coefficients is equal to zero, det A = 0.