Unit One Review
Systems of Linear Equations
Review of terms:
 homogeneous systems of linear equations: systems of equations all whose right
hand side terms are equal to zero.
 non-homogeneous systems of linear equations: systems of equations with at least
one of the right hand side terms is not equal to zero.
 graphical representation of systems of linear equations:
o in 2D: the graph of an equation is a straight line. Three possible scenarios
can occur: either the lines intersect; overlap or parallel.
o in 3D: the graph of an equation is a plane. Three possible scenarios can
occur: either the planes intersect at one point, a line or are parallel
 solution of a system of linear equations: means a set of values of the unknowns
variables that simultaneously satisfy the set of equations. The general solution
consists of a particular solution of the non-homogeneous system plus a solution of
the associated homogeneous system.
 solution set of a system of linear equations: the solution set of a system is either a
singleton (unique solution), infinite (infinitely many solutions) or empty ( no
solution).
 consistent and inconsistent systems of linear equations: consistent systems are
those whose solution set is nonempty while inconsistent systems are those whose
solution set is empty.
 elementary row operations: There are three elementary row operations:
o interchange: two equations (rows)
o multiply: multiple an equation (row) by a non-zero scalar
o replace: replace an equation (row) by its sum with a multiple of another
equation (row).
 equivalent systems: systems that have the same solution set and one is obtained
from the other by applying a sequence of elementary row operations.
 matrix representation of a system of linear equations: A system of linear
equations can be represented as Ax = b where A is the coefficient matrix and
(A | b) is the augmented matrix
 Gauss elimination algorithm: the algorithm reduces a matrix into its row echelon
form
 Gauss-Jordan elimination algorithm: the algorithm reduces a matrix into its
reduced row echelon form
 Back substitution algorithm: the algorithm uses the row echelon form and solves
the system starting with the last nonzero variable say, x k , and then solves for
variables with index less than k.
Review of some facts:
 Every homogeneous system of linear equations is consistent
 Sum and multiple of solutions of homogeneous systems of linear equations are
also solutions of the system

Every system of linear equations Ax = b has either a unique solution, infinitely
many solutions or no solution.
Review Questions:
1. Is every homogeneous system of linear equations consistent? Explain
2. Is every non-homogeneous system of linear equations consistent? Explain
3. If all the rows of an nxn augmented matrix A in the row echelon form are nonzero, describe the solution set of the system associated with the matrix A.
4. If all the rows of an nxn augmented matrix A in the row echelon form are nonzero, describe the solution set of the linear system associated with the matrix A.
5. If at least one of the rows of an nxn augmented matrix A in the echelon form is
zero, describe the solution set of the linear system associated with the matrix A.
6. If the augmented matrix of a system of linear equations in reduced form is
1 2 -1 5


0 1 2 0




0
0
0
a


for what value of a does the system have infinitely many solutions? How many
free variables are there? Write the general solution of the system. Specify the
particular solution.
7. If the augmented matrix of a system of linear equations in reduced form is
1 2 -1 5 4


0 1 2 0 0




0
0
0
0
0


How many free variables are there? Write the general solution of the system.
Specify the particular solution.
8. If the augmented matrix of a system of linear equations in reduced form is
a
1 2 4



0 1 -2
b  2 a 

0 0 1 c 3 a  2 b 


does this system have a solution for any choice of a, b, and c? How many
solutions? Write the general solution of the system. Specify the particular
solution.
9. If the augmented matrix of a system of linear equations in reduced form is
1

0

0

2
1
0
4
-2
0
a


b  2 a 
c 3 a  2 b 
Does this system have a solution for any choice of a, b, and c? How many
solutions? Write the general solution of the system. Specify the particular
solution.
10. Show that if a homogeneous system has one non trivial solution, then is has
infinitely many solutions.