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Math 268, Homework #1, Spring 2012 Name _______________________________________________
Instructions: Write your work up neatly and attach to this page. Record your final answers (only)
directly on this page. Use exact values unless specifically asked to round.
1. Solve the system of equations using elementary row operations on the augmented matrix for
the system. State your solution in vector form, or state that the solution is inconsistent. If the
solution is consistent, state whether it is dependent or independent.
a.
3x1  6 x2  3

5 x1  7 x2  10
b.
 6 x3  8
2 x1

x2  2 x3  3

3x  6 x  2 x  4
2
3
 1
c.
 4 x4  10
2 x1

3x2  3x3
0


x3  4 x4  1

3x1  2 x2  3x3  x4  5
2. Determine the value of any variables to make the system consistent.
1
2

 4
b. 
2
a.
c.
5
8 6 
12 h 
6 3
h
 1 4 7 g 
 0 3 5 h 


 2 5 9 k 
3. For each statement below determine if it is true or false. If the statement is false, briefly explain
why it is false and give the true statement.
a. Every elementary row operation is reversible.
b. A 5x6 matrix has six rows.
c. The solution set of a linear system involving variables x1 ,..., xn is a list of numbers ( s1 ,..., sn )
that makes each equation in the system a true statement when the values s1 ,..., sn are
substituted for x1 ,..., xn respectively.
d. Two fundamental questions about a linear system involve existence and uniqueness.
e. Two equivalent linear systems can have different solution sets.
4. Row reduce the following matrices to echelon and reduced echelon form (report both versions).
Explain why echelon form is not unique but reduced echelon form is.
1 2 4 5 
2 4 5 4
a.


 4 5 4 2 
1
0
b. 
0

0
0 9
1 3
0 0
0 0
0 4
0 1
1 7 

0 1
5. Give 5 examples of a 4x5 matrix in echelon form. Use only 0, 1, * to fill your matrix, with 1
marking the pivot points. For each matrix, indicate whether the solution is consistent or
inconsistent; and if consistent whether it is independent or dependent. For dependent systems,
also list the number of free variables.
6. For each statement below determine if it is true or false. If the statement is false, briefly explain
why it is false and give the true statement.
a. The row reduction algorithm applies only to augmented matrices for a linear system.
b. Finding a parametric description of the solution set of a linear system is the same as solving
the system.
c. If one row in an echelon form of an augmented matrix is 0 0 0 5 0 , then the
associated linear system is inconsistent.
d. If every column of an augmented matrix has a pivot, then the system is consistent.
e. The pivot positions in a matrix depend on whether row interchanges take place.
f.
Whenever a system has free variables, the solution set contains many solutions.
7. Find the interpolating polynomial p(t )  a0  a1t  a2t 2  a3t 3 for the data (1,7), (2,17), (3,31),
(4, 65). That is, plug in the points to obtain a system of 4 equations, and then solve the system
for the coefficients.
8. Rewrite the systems in #1 in matrix form, and in vector form. Give the dimensions of the span
of the column vectors in each case.
1
 4 
3
0


9. List 5 vectors in the span of
and   .
 2 
1
 
 
0
 1
10. For each statement below determine if it is true or false. If the statement is false, briefly explain
why it is false and give the true statement.
 2 
 5 
and

 2  lie on a line through the origin.
5
 
a. The points in the plane corresponding to 
b. An example of a linear combination of vectors v1 and v2 is the vector
c. Any list of 5 real numbers is a vector in

5
1
v1 .
2
.

d. Asking whether b is in the span a1 , a2 , a3 is the same thing as asking whether the linear
system corresponding to an augmented matrix  a1

a2
a3 b  has a solution.
e. The equation Ax  b is referred to as a vector equation.
f.
The equation Ax  b is consistent if the augmented matrix  A b has a pivot in every
row.
g. If the columns of an mxn matrix A span
each b in m .
m
11. Determine if the columns of each matrix spans
 4 5
 3 7
a. 
 5 6

9 1
1
4
1
10
8
2 
4

7
, then the equation Ax  b is consistent for
4
.
 5 11 6 7 12 
 7 3 4 6 9 

b. 
 11 5 6 9 3


 3 4 7 2 7 
12. Determine if the homogeneous systems have a nontrivial solution. If it does, write the solution
in parametric form.
a.
2 x1  5 x2  8 x3  0

3x1  4 x2  2 x3  0
 x1  2 x2  3x3  0

b. 
x2  2 x3  0
2 x  4 x  9 x  0
2
3
 1
c.
 4 x4  x5  0
2 x1

3x2  3x3
 x5  0


x3  4 x4  6 x5  0

3x1  2 x2  3x3  x4  2 x5  0
13. For each statement below determine if it is true or false. If the statement is false, briefly explain
why it is false and give the true statement.
a. A homogeneous equation is always consistent.
b. The homogeneous equation Ax  0 has the trivial solution if and only if the equation has a
least one free variable.
c. The equation x  p  tv describes a line through v parallel to p .
d. If Ax  b is consistent, then the solution of Ax  b is obtained by translating the solution
set of Ax  0 .
2
 
14. Construct a 3x3 nonzero matrix A such that the vector 1 is a solution of Ax  0 .
 
 1 
15. Describe the possible echelon forms of the matrices below using 0, 1 for the pivot and * for all
other entries.
a. A is a 2x2 matrix with linearly independent columns.
b. A is a 4x3 matrix, A   a1



a2


a3  , such that a1 , a2 is linearly independent and a3 is
not in span a1 , a2 .
c. How many pivot columns must a 6x4 matrix have if its columns are independent? Why?