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Math 110—Linear Algebra
Fall 2009, Haiman
Problem Set 3
Due Monday, Sept. 21 at the beginning of lecture.
1.
Show that if vectors
v
1
,...,v
k
in a vector space
V
have the properties that
v
1
6
= 0, and
each
v
i
is not in the span of the preceding ones, then the vectors are linearly independent.
Conversely, show that if
v
1
,...,v
k
is an ordered list of linearly independent vectors, then it
has the above properties.
2.
(a) Find a formula for the number
Q
(
n,k
) of (ordered) sequences (
v
1
,v
2
,...,v
k
) of linearly
independent vectors in
V
, where
V
is a vector space of dimension
n
over
F
2
, and
k
≤
n
.
[Hint: Use the previous problem and Problem Set 2, Problem 5.]
(b) Prove that the number of
k
dimensional subspaces of (
F
2
)
n
is given by
Q
(
n,k
)
/Q
(
k,k
), for
k
≤
n
.
(c) Calculate the number of 5dimensional subspaces of (
F
2
)
10
.
3.
Let
c
1
,...,c
n
be distinct elements of a ﬁeld
F
. Deﬁne the function
E
:
P
m
(
F
)
→
F
n
by
E
(
f
(
x
)) = (
f
(
c
1
)
,...,f
(
c
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This note was uploaded on 08/16/2010 for the course MATH 110 taught by Professor Gurevitch during the Fall '08 term at Berkeley.
 Fall '08
 GUREVITCH
 Linear Algebra, Algebra, Vectors, Vector Space

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