2012 Peking University grad school entrance exam (Higher Algebra)

From math.SE:

1) Let $\xi_1, \xi_2, \ldots, \xi_n$ be all the roots of a polynomial $g(x)$ (defined over the complex field) with rational coefficients. Suppose $f(x)$ is an arbitrary polynomial with rational coefficients. Is $\prod_{i=1}^n f(\xi_i)$ necessarily a rational number? Prove your assertion.

2) Show that the following determinant is nonzero:

$$
\left|\begin{matrix}
1 & 2 & 3 & \ldots & \ldots & \ldots & 2010 & 2011\\
2^2 & 3^2 & 4^2 & \ldots & \ldots & \ldots & 2011^2 & \color{red}{2012^2}\\
3^3 & 4^3 & 5^3 & \ldots & \ldots & \ldots & \color{red}{2012^3} & 2012^3\\
\vdots\\(k-1)^{k-1} & k^{k-1} & (k+1)^{k-1} & \ldots & 2011^{k-1} & \color{red}{2012^{k-1}} & \ldots & 2012^{k-1}\\
k^k & (k+1)^k & (k+2)^k & \ldots & \color{red}{2012^k} & 2012^k & \ldots & 2012^k\\
\vdots\\
2010^{2010} & 2011^{2010} & \color{red}{2012^{2010}} & \ldots & \ldots & \ldots & \ldots & 2012^{2010}\\
2011^{2011} & \color{red}{2012^{2011}} & \ldots & \ldots & \ldots & \ldots & \ldots & 2012^{2011}\\
\end{matrix} \right|.
$$

3) An order $n$ matrix $A$ has exactly one nonzero entry on each row and each column, whose value is either $1$ or $-1$. Show that $A^k=I$ for some positive integer $k$.

4) Define the "product" of two $n\times n$ matrices $A=(a_{ij}), B=(b_{ij})$ as

$$
A\circ B = \left(\begin{matrix}
a_{11}b_{11} & a_{12}b_{12} & \ldots & a_{1n}b_{1n}\\
a_{21}b_{21} & a_{22}b_{22} & \ldots & a_{2n}b_{2n}\\
\vdots\\a_{n1}b_{n1} & a_{n2}b_{n2} & \ldots & a_{nn}b_{nn}
\end{matrix}\right).
$$
If $A$ and $B$ are positive definite, show that $\mathrm{rank}\,A\circ B\le(\mathrm{rank}\,A)(\mathrm{rank}\,B)$.

5) Suppose $f_1,f_2,\ldots,f_{2012}$ are 2012 different linear transformations on a vector space $V$. Does there exist a vector $\alpha\in V$ such that $f_1(\alpha),f_2(\alpha),\ldots,f_{2012}(\alpha)$ are mutually different? Prove your assertion.

6) Suppose $A$ and $B$ are positive definite matrices of order $n$. Prove that they can be simultaneously congruence-diagonalized by some invertible matrix $T$.

7) Let $f(\alpha,\beta)=g_1(\alpha)g_2(\beta)$ be a symmetric bilinear form on a Euclidean vector space $V$ over a field $P$. Show that there exist some linear function $h(x)$ and some $k\in P$ such that $f(\alpha,\beta)=kh(\alpha)h(\beta)$.

8) For any $n$-dimensional Euclidean vector space $V$, show that there are at most $n+1$ vectors such that the angle between any two of them is obtuse.

11) Given that the following linear transormation is a rotation in $\mathbb{R}^3$. Find the rotation axis and the angle of rotation.
$$
\left(\begin{matrix}x'\\ y'\\ z'\end{matrix}\right)
=\left(\begin{matrix}
\frac{11}{15}&\frac{ 4}{15}&\frac{ 2}{ 3}\\
\frac{ 4}{15}&\frac{13}{15}&-\frac{1}{ 3}\\
-\frac{2}{3}&\frac{1}{3}&\frac{2}{ 3}.
\end{matrix}\right)
\left(\begin{matrix}x\\ y\\ z\end{matrix}\right)
$$