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?

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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 .

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 .

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 .

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 .