Linear Algebra - Fall 2000
TEST #1
Name:
h such that the matrix is the augmented matrix of a consistent linear
 2 6 3
system. 
 (8 pt.)
 4 12 h 
1.
Determine the value of
2.
Solve the following system of equations and give the general solution if it exists. Do not leave it in
matrix form. (8 pt.)
x  y  2z  6
3y  z  w  3
2 x  y  2 z  w  9
3.
Prove or disprove whether the following matrices are equivalent or not. Explain how you came up with
your answer. (8 pt.)
 1 2 3  3 2 5 
 0 1 1  5 4 9 

 

4.
The average of the temperatures for the cities of Chicago, Paris and Platteville was 30 during a given
day last March. The temperature in Paris was 9 higher than the average temperatures of the other two
cities. The temperature in Platteville was 9 lower than the average temperature of the other two cities.
What was the temperature in each on of the cities? Give the system of linear equations that you use to
determine your answer. (10 pt.)
5.
Indicate whether the following statement is true or false. If it is false correct the statement. (4 pt. each)
a) A linear system which is consistent has a unique solution if and only if all the variables are free.
b) If there is one row of the coefficient matrix of the form
0,0,
,0 : c then the linear system
associated with the matrix is inconsistent.
c)
The solution set of the linear system with augmented matrix
solution set of the equation
d) The equation
e)
x1 a1  x2 a2 
 a1 , a2 ,

, an , b  is the same as the
 xn an  b .
Ax  b has a solution if and only if  A : b  has a pivot position in every row.
A non-homogeneous equation always is consistent.
1 0 0 


4
f) The column vectors of 0 1 0 span  .


0 0 1 
6.
Give the three equations that would find the temperatures at
x1 , x2 , x3 shown in the figure if the
temperature of each interior point is the average of its four neighboring points. Give the three
equations you used to get you solution. (8 pt.)
1
2
2
1
x1
x2
2
x3
2
1
1
3
 2




 
7. Given that v1  3 , v2  8 , v3  3 . This problem is designed so that you should not be
 
 
 
3
7 
 h 
able to use your calculator. Show work!
a)
For what values of
b) For what values of


h is v3 in Span v1 , v2 . (8 pt.)


h is v1 , v2 , v3 linearly independent? (8 pt.)
 3 1 1 0 


8. Determine if the columns of the matrix 2 1 0 1 are linearly dependent or dependent. If


 4 1 1 1 
the columns are dependent find a vector equation which would prove this fact. (The equation would
have to use the columns of the matrix  a1 , a2 , a3 , a4  as vectors.) (7 pt.)


1 
0


 
9. Describe geometrically in  the following sets when v1  0 , v2  0 (Your description
 
 
0
1 
3
should tell me exactly what the span of each set represents.)
a)
Span v1 (3 pt.)
b)
Span v1 , v2  (3 pt.)