Quiz: Possibilities For the Solution Set of a Homogeneous System of Linear Equations
Problem 93
4 multiple choice questions about possibilities for the solution set of a homogeneous system of linear equations.
The solutions will be given after completing all problems.
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Question 1 of 4
1. Question
True or False: A homogeneous system has always infinitely many solutions.
Correct
Good!! A homogenous solution could have only one solution.
Incorrect
A homogenous solution could have only one solution.

Question 2 of 4
2. Question
Determine all possibilities for the solution set of the system.
A homogenius system of 4 equations in 3 unknowns.
Correct
Good!! Since the homogeneous system is always consistent, the rank $r$ of the augmented matrix is either $r=0, 1, 2, 3$.
Then the number of free variables is $nr$, where $n=3$ in this case. Thus when $r=3$, there is no free variable, hence the system has a unique solution. If $r<3$, then the system has at least one free variable, hence there are infinitely many solutions.Incorrect
Since the augmented system is always consistent, the rank $r$ of the augmented matrix is either $r=0, 1, 2, 3$.
Then the number of free variables is $nr$, where $n=3$ in this case. Thus when $r=3$, there is no free variable, hence the system has a unique solution. If $r<3$, then the system has at least one free variable, hence there are infinitely many solutions. 
Question 3 of 4
3. Question
Determine all possibilities for the solution set of the system.
A homogenius system of 5 equations in 7 unknowns.
Correct
Good!! For a homogenous system, if there are more unknowns than equations, then the system has always infinitely many solutions.
Incorrect
For a homogenous system, if there are more unknowns than equations, then the system has always infinitely many solutions.

Question 4 of 4
4. Question
Determine all possibilities for the solution set of the system.
\begin{align*}
a_{11}x_1 + a_{12}x_2 +a_{13} x_3=0 \\
a_{21}x_1 + a_{22} x_2 + a_{23} x_3 =0 \\
a_{31}x_1 + a_{32}x_2 +a_{33}x_3 =0
\end{align*}This system has a solution $x_1=2, x_2=5, x_3=7$.
Correct
Good!! Note that the system is homogeneous. A homogeneous system always has a zero solution, hence it is consistent. For this problem, we have another solution $x_1=2, x_2=5, x_3=7$.
This implies that the system has at least two solutions. Thus it must have infinitely many solutions.Incorrect
Note that the system is homogeneous. A homogeneous system always has a zero solution, hence it is consistent. For this problem, we have another solution $x_1=2, x_2=5, x_3=7$.
This implies that the system has at least two solutions. Thus it must have infinitely many solutions.
(The Ohio State University, Linear Algebra Exam)
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