Math 115AH Homework # 2
(additional problems)
Problem 1
Let
be a vector space which is spanned by a finite set
consisting
of
elements. Show that
is finite-dimensional with dimension
less then or equal to
. (You should not assume from the beginning that
is finite-dimensional!)
Problem 2
a) Let
. Prove that
these functions are linearly independent in the space
of all complex-valued functions on the real line.
b) Let
.
Find an invertible
matrix such that
for all
.
Problem 3
Let
be a
-dimensional vector space, and
be its
-dimensional
subspace. Show that there is a
-dimensional subspace
of
such that
.
Problem 4
Let
,
be two
vector spaces, both of dimension
. Construct an explicit isomorphism
from
to
.
Problem 5
a) Prove that a linear map
is an isomorphism iff it sends a
basis of
to a basis of
.
b) Let
and
be vector spaces, and let
.
Let
be a basis in
and
be an arbitrary set of vectors in
. Prove (by giving an explicit construction)
that there exists a linear transformation
such that
for all
. Is such a transformation unique?
Problem 6
(Annihilators) Let
be a subset of a vector space
.
- Prove that its annihilator
is a subspace of .
- Prove that if
is a subspace and , then there
exists
such that
.
- Let be finite-dimensional and
be the isomorphism
defined by
, where
for all
.
Prove that
. In particular, if
is a subspace, conclude that
(i.e., and
can be identified by the natural isomorphism between and ).
- Let and be subspaces. Show that
and
. (Hint: first prove
that
implies that
).
Math 115AH Homework # 2
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