Math 4310
Homework 10 Solutions
1. Let F = C, and let m, n N with 0 < m n. Prove that there exists a polynomial p(x) F[x]
of degree n with exactly m distinct roots. What goes wrong if F is a nite eld?
S OLUTION .
One can choose m distinct complex numbers
Math 4310
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Homework 10
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Due 11/30/2012
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Math 4310
Homework 9
Due 11/21/2012
Exercises.
1. In R4 , let
U = S(1, 1, 0, 0), (1, 1, 1, 2).
Find u U such that |u (1, 2, 3, 4)| is as small as possible.
Answer. We want to nd the orthogonal projection of (1, 2, 3, 4) onto U. To do this, we rst
nd an or
Math 4310
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Homework 9
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Due 11/21/2012
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Math 4310
Homework 8 Solutions
1. Suppose that u, v V are in an inner product space, and
|u| = 3, |u + v| = 4, and |u v| = 6.
What number must |v| equal?
S OLUTION . The computation done in exercise 2 shows that
u+v
Hence 2 v
2
2
2
+ uv
= 42 + 62 2 32 = 3
Math 4310
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Homework 8
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Due 10/31/2012
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Math 4310
Homework 7 Solutions
Due 10/24/2012
Exercises.
1. Consider the elementary matrices dened in class (and in the text in 12, Example C). These
are denoted Pi,j , Bi,j (), and Di () for = 0.
1
(a) Pi,j1 = Pi,j , Bi,j ()1 = Bi,j (), and Di ()1 = Di (
Math 4310
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Homework 7
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Due 10/24/2012
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Math 4310
Homework 6 Solutions
1. Let S and T be subspaces of Fn which are represented as the solution spaces of homogeneous
equations,
S = So (a1
x = 0, . . . , as
x = 0) and T = So (b1
x = 0, . . . , bt
x = 0).
(a) Prove that S T is the solution space
S
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Homework 6
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Due 10/17/2012
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Math 4310
Homework 5
Due 9/28/12 (!)
Exercises.
1. Let f1 , f2 and f3 be vectors in the vector space F un(R, R) = cfw_f : R R over R.
A. For three distinct real numbers x1 , x2 and x3 , dene a matrix
f1 (x1 ) f1 (x2 ) f1 (x3 )
[fi (xj )] = f2 (x1 ) f2 (x2
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Math 4310
Homework 4 Solutions
1. Which of the following sets of vectors in the indicated vector spaces are linearly independent? Please justify your responses.
A. (1, 1), (2, 1), and (1, 2) in R2 as a vector space over R.
S OLUTION . R2 is a vector space
Math 4310
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Homework 4
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Due 9/19/12
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Math 4310
Homework 3
Due 9/12/12
Exercises.
1. (Curtis, [1, p. 26 #8]) A quadrilateral is a set of four distinct points cfw_A, B, C, D in R2 . The
sides of the quadrilateral are the line segments AB, BC, CD and DA.
Claim. The midpoints of the sides of a q
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Math 4310
Homework 2 Solutions
1. First let us prove that for any integer n 4,
n2 2n .
Proof. We proceed by induction on n. First, we prove a base case: when n = 4,
42 = 16 24 .
We now assume that n2 2n for some integer n 4 and show that (n + 1)2 2n+1 . B
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Homework 2
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Due 9/5/12
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Math 4310
Homework 1 Solutions
1.
(a) Claim. For any integer n 1,
n
12 + 22 + + n2 =
k2 =
k =1
n(n + 1)(2n + 1)
.
6
Proof. We proceed by induction on n. First, we consider the case when n = 1. We have
1
k2 = 12 = 1 =
k =1
1(1 + 1)(2 1 + 1)
6
as desired.
W
Math 4310
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Homework 1
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Due 8/29/12
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