1
Math E21b – Spring 2011 – Homework #1
Problems due in class on Thurs, Feb 3
:
Section 1.1
:
In exercise 11, find all solutions of the linear systems. Represent your solutions graphically, as intersections of
lines in the
xy
plane.
11.
22
351
7
xy
In Exercise 16, find all solutions of the linear system. Describe your solution in terms of intersecting planes.
You need not sketch these planes.
16.
40
41
370
72
21
30
z
z
z
Note
: In understanding the relationship between linear equations and the geometry of planes in
R
3
,
it is helpful to understand the
dot product
, a topic covered in Math E21a. There’s a brief
summary of vectors, the dot product and the cross product in Appendix A of the Bretscher text,
but it’s best to consult any standard multivariable calculus text for a more complete treatment of
these topics. A supplement is posted on the course website. We can also go over this in more
detail outside of class.
17. Find all solutions of the linear system
2
35
x
ya
x
yb
, where
a
and
b
are arbitrary constants.
25. Consider the linear system
2
3
5
13
18
25
xyz
z
x
yz
k
, where
k
is an arbitrary number.
a. For which value(s) of
k
does this system have one or infinitely many solutions?
b. For each value of
k
you found in part a, how many solutions does the system have?
c. Find all solutions for each value of
k
.
29. Find the polynomial of degree 2 [a polynomial of the form
2
()
f
ta
b
t
c
t
] whose graph goes through
the points
(1, 1)
,
(2,3) , and (3,13) . Sketch the graph of this polynomial.
Section 1.2
:
In exercise 9 and 11, find all solutions of the equations with paper and pencil using GaussJordan elimination.
Show all your work.
9.
45
6
12
5
6
123
5
6
20
2
xx
x
x
x
xxx
x
x
.
11.
13
4
234
1234
24
8
36
3468
0
34
1
2
x
xxxx
20. We say that two
n
m
matrices in reduced rowechelon form are of the same type if they contain the same
number of leading 1’s in the same positions. For example,
120
001
and
130
are of the same type. How many types of 2
2 matrices in reduced rowechelon form are there?
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 Spring '10
 DR.RAHULA
 Economics, ij, rowechelon form, Inputoutput model

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