Page 73 - Advanced Linear Algebra

Vector Spaces 57

14. Let be a finite-dimensional vector space over an infinite field . Prove
-
=
are subspaces of of equal dimension, then there is a
that if :Á Ã Á : =
=
;
subspace of for which = ~ : l ; for all ~ ÁÃÁ . In other words,

; : is a common complement of the subspaces .
15. Prove that the vector space of all continuous functions from to is
9
s
s
infinite-dimensional.
16. Show that Theorem 1.2 need not hold if the base field is finite.
-
:
17. Let be a subspace of . The set # = b : ~ ¸ # b “  : ¹ is called an
affine subspace of .
=
)
a Under what conditions is an affine subspace of a subspace of ?
=
=
b Show that any two affine subspaces of the form #b: and $b: are
)
either equal or disjoint.
-
=
18. If and > are vector spaces over for which = ~ (( ( > ( , then does it
follow that dim²= ³ ~ dim²> ? ³

:
19. Let = be an -dimensional real vector space and suppose that is a
subspace of with dim ² : ³ ~ c . Define an equivalence relation – on
=
the set =± : by # – $ if the “line segment”
3²#Á $³ ~ ¸ # b ² c ³$ “   ¹
has the property that 3²#Á $³ q : ~ J . Prove that – is an equivalence
relation and that it has exactly two equivalence classes.
20. Let be a field. A subfield of is a subset 2 - of that is a field in its
-
-
own right using the same operations as defined on .
-
)
a Show that is a vector space over any subfield of .
-
2
-
)
b Suppose that is an -dimensional vector space over a subfield of
2
-

-
- =. If is an -dimensional vector space over , show that is also a
=
vector space over 2 . What is the dimension of as a vector space
=
over ?
2
21. Let be a finite field of size and let be an -dimensional vector space

-
=

over - . The purpose of this exercise is to show that the number of
subspaces of of dimension is

=

² c ³Ä² c ³
45 ~
² c ³Ä² c ³² c c ³Ä² c ³

are called Gaussian coefficients and have properties
The expressions ²³

similar to those of the binomial coefficients. Let :² Á ³ be the number of
=-dimensional subspaces of .
a Let 5² Á ³ be the number of -tuples of linearly independent vectors
)

²# ÁÃÁ# ³ in . Show that
=

5² Á ³ ~ ² c ³² c ³Ä² c c ³
)
)
b Now, each of the -tuples in a can be obtained by first choosing a

subspace of of dimension and then selecting the vectors from this

=
subspace. Show that for any -dimensional subspace of , the number
=