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.
Add to solve later
Sponsored Links
Quizsummary
0 of 4 questions completed
Questions:
 1
 2
 3
 4
Information
You must fill out this field. 
Answer the following questions.
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading...
You must sign in or sign up to start the quiz.
You have to finish following quiz, to start this quiz:
Results
0 of 4 questions answered correctly
Your time:
Time has elapsed
You have reached 0 of 0 points, (0)
Average score 

Your score 

Categories
 Not categorized 0%

Click View questions button below to see the answers.
 1
 2
 3
 4
 Answered
 Review

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)
Add to solve later
Sponsored Links
More from my site
 Summary: Possibilities for the Solution Set of a System of Linear Equations In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems. Determine all possibilities for the solution set of the system of linear equations described below. (a) A homogeneous system of $3$ […]
 The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns Determine all possibilities for the number of solutions of each of the system of linear equations described below. (a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=3, x_3=1$ as a solution. (b) A homogeneous system of $5$ equations in $4$ unknowns and the […]
 Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) \[\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right. \] where $a,b,c, d$ […]
 Quiz 1. GaussJordan Elimination / Homogeneous System. Math 2568 Spring 2017. (a) Solve the following system by transforming the augmented matrix to reduced echelon form (GaussJordan elimination). Indicate the elementary row operations you performed. […]
 If the Augmented Matrix is RowEquivalent to the Identity Matrix, is the System Consistent? Consider the following system of linear equations: \begin{align*} ax_1+bx_2 &=c\\ dx_1+ex_2 &=f\\ gx_1+hx_2 &=i. \end{align*} (a) Write down the augmented matrix. (b) Suppose that the augmented matrix is row equivalent to the identity matrix. Is the system consistent? […]
 Linear Algebra Midterm 1 at the Ohio State University (1/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 1 and contains the […]
 A Condition that a Linear System has Nontrivial Solutions For what value(s) of $a$ does the system have nontrivial solutions? \begin{align*} &x_1+2x_2+x_3=0\\ &x_1x_2+x_3=0\\ & 3x_1+4x_2+ax_3=0. \end{align*} Solution. First note that the system is homogeneous and hence it is consistent. Thus if the system has a nontrivial […]
 Find Values of $a$ so that Augmented Matrix Represents a Consistent System Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations. \[A= \left[\begin{array}{rrrr} 1 & 2 & 3 & 4 \\ 2 &1 & 2 & a^2 \\ 1 & 7 & 11 & a \end{array} \right],\] where $a$ is a real number. Determine all the […]