Let and be complete flags over an -dimensional vector space . Show that there is a set of lines compatible with the two flags, in the sense that there exists a permutation such that and .

# linear algebra

# Not everything is (only) abstract nonsense

Prove that is surjective for every integer but is not necessarily surjective.

# An exercise in Linear Algebra

Let be a field of characteristic and be the -vector space of homogeneous degree polynomials in variables. Show that the linear span of the set of -th powers of linear polynomials is the whole space of homogeneous degree polynomials.

In coordinate-free terms: if is a finite-dimensional -vector space then spans .

# [Some punny title about determinants and fixed points]

Prove the following identity

the sum running through and the matrix being of size .

# Its hip to be skew

Show that the determinant of a skew symmetric matrix is never reducible when viewed as a polynomial in variables.

# The order of an automorphism

Let . What is the maximum order can have?

# Keeping it real

Let be a matrix with a real, simple eigenvalue. Prove that there exists a neighborhood of in which all matrices have at least one real eigenvalue.