Question 31
Let $a$ and $b$ be different integers of $0$, $1$ and $-1$, and consider the matrix $$
\left(
\begin{array}{ccccc}
a+b & a+b^{2} & a+b^{3} & \ldots & a+b^{m} \\
a^{2}+b & a^{2}+b^{2} & a^{2}+b^{3} & \ldots & a^{2}+b^{m} \\
a^{3}+b & a^{3}+b^{2} & a^{3}+b^{3} & \ldots & a^{3}+b^{m} \\
\vdots & \vdots & \vdots & & \vdots \\
a^{n}+b & a^{n}+b^{2} & a^{n}+b^{3} & \ldots & a^{n}+b^{m}
\end{array}
\right)
$$ Determine a subset $S$ of rows of that matrix, as little as possible, such that any other row can be expressed as the sum of the rows of $S$ multiplied by integers appropriate (ie, as a linear combination with integer coefficients of the rows of $S$). make these linear combinations explicit.
\left(
\begin{array}{ccccc}
a+b & a+b^{2} & a+b^{3} & \ldots & a+b^{m} \\
a^{2}+b & a^{2}+b^{2} & a^{2}+b^{3} & \ldots & a^{2}+b^{m} \\
a^{3}+b & a^{3}+b^{2} & a^{3}+b^{3} & \ldots & a^{3}+b^{m} \\
\vdots & \vdots & \vdots & & \vdots \\
a^{n}+b & a^{n}+b^{2} & a^{n}+b^{3} & \ldots & a^{n}+b^{m}
\end{array}
\right)
$$ Determine a subset $S$ of rows of that matrix, as little as possible, such that any other row can be expressed as the sum of the rows of $S$ multiplied by integers appropriate (ie, as a linear combination with integer coefficients of the rows of $S$). make these linear combinations explicit.