Page 72 - Advanced Linear Algebra
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56 Advanced Linear Algebra
2. Prove Theorem 1.3.
)
3. a Find an abelian group and a field for which is a vector space
=
-
=
over in at least two different ways, that is, there are two different
-
definitions of scalar multiplication making a vector space over .
-
=
b Find a vector space = over - and a subset of = : that is 1 a
)
(
)
subspace of and 2 a vector space using operations that differ from
()
=
those of .
=
:
4. Suppose that is a vector space with basis 8 ~ ¸ 0 ¹ and is a
=
=
subspace of . Let ) ¸ Á Ã Á ) ¹ be a partition of . Then is it true that
8
:~ ²: q º) »³
~
What if : q º) » £ ¸ ¹ for all ?
5. Prove Theorem 1.8.
I
6. Let :Á ;Á < ²= ³ . Show that if < : , then
: q ²; b <³ ~ ²: q ;³ b <
This is called the modular law for the lattice I²= ³ .
7. For what vector spaces does the distributive law of subspaces
: q ²; b <³ ~ ²: q ;³ b ²: q <³
hold?
8. A vector # ~ ² ÁÃÁ ³ s is called strongly positive if for all
~ Á Ã Á .
)
a Suppose that is strongly positive. Show that any vector that is “close
#
(
enough” to is also strongly positive. Formulate carefully what “close
#
enough” should mean.)
b Prove that if a subspace of s contains a strongly positive vector,
)
:
then has a basis of strongly positive vectors.
:
9. Let 4 be an d matrix whose rows are linearly independent. Suppose
*
of 4 span the column space of 4 . Let be
that the columns ÁÃÁ
the matrix obtained from 4 by deleting all columns except ÁÃÁ .
Show that the rows of are also linearly independent.
*
10. Prove that the first two statements in Theorem 1.7 are equivalent.
11. Show that if is a subspace of a vector space , then dim ²:³ dim ²= . ³
:
=
Furthermore, if dim²:³ ~ dim²= ³ B then : ~ = . Give an example to
show that the finiteness is required in the second statement.
12. Let dim²= ³ B and suppose that = ~ <l : ~ <l : . What can you
say about the relationship between : and : ? What can you say if
?
: :
13. What is the relationship between :l ; and ; l : ? Is the direct sum
operation commutative? Formulate and prove a similar statement
concerning associativity. Is there an “identity” for direct sum? What about
“negatives”?
