Quiz 05
ID#:
Name:
1. The echelon form of an augmented matrix of a system of linear equations is
a1
1 0 1


given by A  0 1 2
a 2  3a1 

0 0 0 a3  2a 2  a1 
Then the system has infinitely many solutions if:
a. a1 , a 2 a3 arbitrary.
b. a1  a2  a3  0
c. a3  2a2  a1 .
d. a3  2a2  a1
2. Any homogeneous system of linear equations:
a. Has at least one solution.
b. Has an empty solution set.
c. Has only the trivial solution.
d. Has only non trivial solutions.
3. Any nonhomogeneous system of linear equations:
e. Has a non-empty solution set.
f. Has an empty solution set.
g. Has only the trivial solution.
h. Non of the above.
4. If one is solving two linear equations involving three unknowns, what
happens?
(a) Usually there will be no solution, but occasionally there will be one or more
solutions.
(b) There will never be a solution.
(c) There will always be a solution.
(d) There will always be infinitely many solutions.
(e) Usually there will be one solution, but occasionally there will be no
solutions or infinitely many solutions
(f) Usually there will be infinitely many solutions, but occasionally there will be
one or no solutions.
(g) Usually there will be infinitely many solutions, but occasionally there will be
no solutions.
(h) There will always be exactly one solution.
5. If one is solving three linear equations involving three unknowns, what
happens?
(a) Usually there will be no solution, but occasionally there will be one or more
solutions.
(b) There will never be a solution.
(c) There will always be a solution.
(d) There will always be infinitely many solutions.
(e) Usually there will be infinitely many solutions, but occasionally there will be
one or no solutions.
(f) Usually there will be one solution, but occasionally there will be no
solutions or infinitely many solutions.
(g) Anything can happen.
(h) There will always be exactly one solution.
6. What is the complete relationship between homogeneous linear systems of
equations, and the zero solution (all unknowns equal to zero)?
(a) The zero solution is always a solution to both homogeneous and
inhomogeneous linear systems.
(b) The zero solution is always a solution to homogeneous linear systems, but
could also be a solution to inhomogeneous linear systems.
(c) The zero solution is never a solution to inhomogeneous linear systems,
and may or may not be a solution to homogeneous linear systems.
(d) The zero solution can be a solution to both homogeneous and
inhomogeneous linear systems, but only if the equations are solvable.
(e) If a solution to a homogeneous linear system exists at all, it will be the zero
solution.
(f) If a solution to a homogeneous linear system exists at all, then the zero
solution will be a solution.
(g) The zero solution is always a solution to homogeneous linear systems, and
never a solution to inhomogeneous linear systems.
7. If a linear system has four unknowns and has rank three, then
(a) There are infinitely many solutions.
(b) There are no solutions (system is inconsistent).
(c) There is exactly one solution.
(d) There is either one solution or infinitely many solutions.
(e) There is either one solution or no solution.
(f) There are infinitely many solutions, unless the system is inconsistent, in
which case there are no solutions.
(g) Anything can happen.
8. If a linear system has four unknowns and has rank four, then
(a) There are infinitely many solutions, unless the system is inconsistent, in
which case there are no solutions.
(b) There are infinitely many solutions.
(c) There are no solutions (system is inconsistent).
(d) There is exactly one solution.
(e) There is either one solution or infinitely many solutions.
(f)
There is either one solution or no solution.
(g) Anything can happen.
9.Determine whether the following sets of vectors are independent or
dependent. (4 points for each)
1

(a) 1
1
 
0  
0  
 
0 
 1 
 
(b)  1 
 1 
  
 1 

(c)  1
 2 
 
1
 1
 
 1 
0  
0  
 
1 
 1 

(d) 0
  2
 
1
 1
 
 1 
1 
1 
 
0 
 1 

(e) 0
 2
 
1
 1
 
 1 
2
6 
 
1 
1 
1 
 
0 