University Mathematics Handbook (2015)
X. Algebra
Chapter 7. System of Linear Equations
7.1 Definition and Solution
a. A system of 
linear equations with 
unknowns:
(*)
when 
are unknowns, 
are their coefficients, with the first index 
indicating which equation coefficient 
is, and the second index 
, indicating that 
is the coefficient of 
.
b. 
are the free elements.
c. (*) is a homogeneous system if all its free coefficients equal zero.
d. The coefficient matrix of system (*) is:
e. 
is a vector form of system (*) when
f. The extended matrix of system (*) is
.
g. The numbers are the solution of equation system (*) if, when substituted in each equation, we get an identity.
is the solution vector.
Homogeneous system always has a solution 
.
is called trivial solution.
h. The equation system solution does not change if:
1. Both sides of an equation are multiplied by a non-zero number.
2. A multiple of one equation is added to another equation.
3. The positions of two equations are swapped.
i. A system of equations with unknowns, , has:
1. A unique solution if and only if the rank of matrix 
is equal to the rank of matrix 
and equal to .
2. An infinite number of solutions occurs when 
. In such a case, the system has 
degrees of freedom.
3. There is no solution, when 
.
j. A system of linear equations with unknowns has a unique solution, if and only if 
.
k. Homogeneous system 
has at least one solution, such as the trivial solution.
l. If, in a homogeneous system, number of unknowns is greater than number of equations , then the system has an infinite number of solutions, and its general solution has at least 
degrees of freedom.
m. Cramer theorem: a system with equations and unknowns has a unique solution if and only if 
. The unique solution is expressed by:
when 
is a determinant of the matrix obtained from when column is substituted with 
.
7.2 Null Space
a. The set of solutions of system , when 
is a subspace of 
if and only if 
.
b. The subspace of 
is referred to as the solution space or the null space of .
c. The dimension of the null space of is 
.
d. If 
is a solution of , and 
, x1,…, xk are basis of the null space of , then the general solution of can be expressed as
