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.
Exhibit a curve in such that any -uple of points in its trace is in general position.