Archive
Max Dimension for a linear space of singular matrices
Denote by a vector space of singular (
)
matrices. What is the maximal dimension of such a space.
Denote by a vector space of
matrices with rank smaller or equal to
. What is the maximal dimension of such a space?
Hahn-Banach Theorem (real version)
Suppose is a vector space over
,
has the following properties:
and
.
Let be a subspace of
and
a linear functional such that
.
Then we can find a linear functional such that
and
. Read more…
Fuglede & Putnam Theorem
Fuglede’s Theorem Let be bounded linear operators on a Hilbert space
with
being normal (
). Then
implies
.
Putnam’s Theorem Let be bounded linear operators on a Hilbert space
with
being normal. Then
implies
.
Fixed point for an operator
Suppose is a Hilbert space and
is a linear operator on
with
. Prove that
if and only if
.
Solution: Using the Cauchy-Buniakovski inequality, we get
. Since this implies equality in the C-B inequality, we must have
. We easily find that
. The converse is equivalent to the implication above.
Impossible relation
Prove that we cannot find any two linear continuous maps such that
. Read more…
Disjoint convex sets
Prove that for any disjoint non-void convex sets from a linear space
we can find
convex and disjoint, such that
and
.
Read more…
Linear maximal subspaces & Linear functionals
Take to be a vector space over the field
.
1. Let be a linear functional, which is not identically zero. Then it’s kernel,
, is a linear maximal subspace of
.
2. Conversely, given a maximal subspace of
, there exists a linear functional
such that
.
3. As a consequence of the above prove that if we have two linear functionals (different from zero identity) such that
then there exists
such that
.
As application to the above solve the following:
4. Given linear functionals on
such that
prove that we can find scalars
such that
.
Prove that if are linearly independent linear functionals on
then there exist elements
such that
, where
.