1987 Putnam B5

Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be

a $2n \times n$ matrix of complex numbers such that whenever $(z_1,

z_2, \dots, z_{2n})M = O_n$, with complex$z_i$, not all zero, then at

least one of the $z_i$ is not real. Prove that for arbitrary real

numbers $r_1, r_2, \dots, r_{2n}$, there are complex numbers $w_1,

w_2, \dots, w_n$ such that

$$

\mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array}

\right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_n

\end{array} \right).

$$

(Note: if $C$ is a matrix of complex numbers, $\mathrm{re}(C)$ is the matrix

whose entries are the real parts of the entries of $C$.)

令 $O_n$ 为 $n$ 维向量 $(0,0,\cdots, 0)$。设 $M$ 为

一个 $2n \times n$ 复数矩阵，使得每当 $(z_1,

z_2, \dots, z_{2n})M = O_n$，复数$z_i$，不全为零，则在

$z_i$ 中至少有一个不是真实的。证明对于任意实数

数字 $r_1, r_2, \dots, r_{2n}$, 有复数 $w_1,

w_2, \dots, w_n$ 这样

$$

\mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array}

\right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_n

\end{array}\right)。

$$

(注：如果$C$是复数矩阵，则$\mathrm{re}(C)$是矩阵

其条目是 $C$ 条目的实部。)