Chapter 1: Theory Review: Solutions
Math 308 F
Spring 2015
1. Fill in the blank:
A linear system can have 0, 1 or infinitely many solutions.
2. For each system given below, circle the possibilities for the solution set, and give an
example of a linear system with that solution set.
(a) A linear system with n variables and m equations, with n > m:
i. 0 solutions
ii. exactly 1 solution
iii. infinitely many solutions
If n > m, the system can have 0 or infinitely many solutions (it cannot
have exactly one). For example, the linear system
x1 + x2 + x3 = 0
x1 + x2 + x3 = 1
has 0 solutions and the linear system
x1 + x2 + x3 = 0
x1 − 2x2 + 3x3 = 0
has infinitely many solutions.
(b) A linear system with n variables and m equations, with n ≤ m:
i. 0 solutions
ii. exactly 1 solution
iii. infinitely many solutions
If n ≤ m, the system can have 0, 1, or infinitely many solutions. For
example, the linear system
x1 + x2 = 0
x1 + x2 = 1
has 0 solutions, the linear system
x1 + x2 = 0
x1 − 2x2 = 0
has exactly one solution, and the linear system
x1 + x2 = 2
2x1 + 2x2 = 4
has infinitely many solutions.
(c) A homogeneous linear system with n variables and m equations, with n > m:
i. 0 solutions
ii. exactly 1 solution
iii. infinitely many solutions
If n > m, the homogeneous system must have infinitely many solutions. For example, the linear system
x1 + x2 + x 3 = 0
x1 − 2x2 + 3x3 = 0
has infinitely many solutions.
(d) A homogeneous linear system with n variables and m equations, with n ≤ m:
i. 0 solutions
ii. exactly 1 solution
iii. infinitely many solutions
If n ≤ m, the homogeneous system can have 1 or infinitely many
solutions (it cannot have zero). For example, the linear system
x1 + x2 = 0
x1 − 2x2 = 0
has 1 solution and the linear system
x1 + x2 = 0
4x1 + 4x2 = 0
has infinitely many solutions.
3. For each statement below, determine if it is True or False. Justify your answer.
(a) Any linear system is equivalent to a unique system in echelon form.
FALSE. Any linear system has infinitely many echelon forms. For example,
the matrices
1 1 0
2 2 0
1 0 0
,
,
0 −3 0
0 1 0
0 1 0
are all echelon forms of the system
x1 + x2 = 0
x1 − 2x2 = 0.
(b) Any linear system is equivalent to a unique system in reduced echelon form.
TRUE. This is Theorem 1.6 in section 1.2 of the book.
(c) If a linear system has infinitely many solutions, there must be more equations
than variables.
FALSE. See the example for 2(b)(iii) above for a counter example.
(d) If there are two (distinct) known solutions to a linear system, the system must
have infinitely many more solutions.
TRUE. If there are two different solutions, the system cannot have 0 or 1
solution (it has two!), therefore it must have infinitely many solutions.
(e) If s = (s1 , . . . , sn ) is a solution to a homogeneous system of equations, then
any multiple of s is also a solution.
TRUE. If (s1 , . . . , sn ) is a solution to any homogeneous equation
a1 x1 + a2 x2 + · · · + an xn = 0,
that means
a1 s1 + a2 s2 + · · · + an sn = 0.
But, if r is any constant, (rs1 , . . . , rsn ) is also a solution because, multiplying
the above equation by r, we get
r(a1 s1 + a2 s2 + · · · + an sn ) = 0,
which simplifies to
a1 rs1 + a2 rs2 + · · · + an rsn = 0,
telling us that (rs1 , . . . , rsn ) is also a solution.