# Accute result

Any set of vectors $v_1, \dots, v_n$ in some Euclidean space such that any pair of them lie at an obtuse angle are either linearly independent or have a linear combination with non-negative coefficients equal to zero.

# Lines in flags

Let $F_1 \subseteq \dots \subseteq F_n$ and $F'_1 \subseteq \dots \subseteq F'_n$ be complete flags over an $n$-dimensional vector space $V$. Show that there is a set of $n$ lines $L_1,\dots, L_n$ compatible with the two flags, in the sense that there exists a permutation $\sigma \in S_n$ such that $F_i = L_1 \oplus \dots \oplus L_i$ and $F'_i = L_{\sigma(1)}\oplus \dots \oplus L_{\sigma(i)}$.

# Not everything is (only) abstract nonsense

Prove that $\mathrm{SL}_2(\mathbb{Z})\to\mathrm{SL}_2(\mathbb{Z}/n\mathbb{Z})$ is surjective for every integer $n$ but $\mathrm{GL}_2(\mathbb{Z})\to\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$ is not necessarily surjective.

# An exercise in Linear Algebra

Let $k$ be a field of characteristic $0$ and $k[x_1,\ldots,x_n]^{(d)}$ be the $k$-vector space of homogeneous degree $d$ polynomials in $n$ variables. Show that the linear $k$ span of the set of $d$-th powers of linear polynomials $(a_1 x_1+\ldots+a_n x_n)^d$ is the whole space of homogeneous degree $d$ polynomials.

In coordinate-free terms: if $V$ is a finite-dimensional $k$-vector space then $\{ v^n : v\in V\}$ spans $\mathrm{Sym}^dV$.

# [Some punny title about determinants and fixed points]

Prove the following identity

$\sum_\sigma (-1)^\sigma t^{\mathrm{Fix}\;\sigma} = \det\begin{pmatrix} t & 1 &\cdots& 1& 1\\ 1 & t &\cdots& 1& 1 \\ \vdots & \vdots &\ddots & \vdots & \vdots \\ 1 & 1 &\cdots& t& 1\\ 1 & 1 &\cdots& 1& t \end{pmatrix}$

the sum running through $S_n$ and the matrix being of size $n\times n$.

# Its hip to be skew

Show that the determinant of a skew symmetric $2n\times 2n$ matrix is never reducible when viewed as a polynomial in $\binom {2n} 2$ variables.

# The order of an automorphism

Let $A\in \mathrm{GL}(n, q)$. What is the maximum order $A$ can have?