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 .
Prove that is surjective for every integer but is not necessarily surjective.
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 .
Prove the following identity
the sum running through and the matrix being of size .
Show that the determinant of a skew symmetric matrix is never reducible when viewed as a polynomial in variables.
Let . What is the maximum order can have?
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.