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I've got a large system of equations: $$ \begin{align*} (2^0)^na_n + (2^0)^{n-1}a_{n-1} + \cdots + (2^0)^1a_1 &= 4^0 \\ (2^1)^na_n + (2^1)^{n-1}a_{n-1} + \cdots + (2^1)^1a_1 &= 4^1 \\ \vdots\\ (2^{n-1})^na_n + (2^{n-1})^{n-1}a_{n-1} + \cdots + (2^{n-1})^1a_1 &= 4^{n-1} \\ \end{align*} $$ for $n\geq2$. I'd like to show that the (unique) solution of this system is when $a_2=1$ and all other variables are zero. I'm not very familiar with linear algebra, however I tried putting these equations in matrix form. My attempt involved using induction of variable $n$, and thinking of one matrix as a "sub-matrix" of the next one. Again, due to limited knowledge of linear algebra, I got nowhere with it.

Also, is induction a good idea for such a problem or would you use another method of proof?

Thank you.

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  • $\begingroup$ Vandermonde matrix: en.wikipedia.org/wiki/Vandermonde_matrix. $\endgroup$ – Martín-Blas Pérez Pinilla Oct 4 '14 at 16:20
  • $\begingroup$ It is really useful to know the Vandermonde determinant. There are many techniques to avoid it, but many of them have limitations, particularly over $\mathbb{C}$. $\endgroup$ – Slade Oct 4 '14 at 16:37
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Suppose that $(a_1,\ldots,a_n)$ is a solution of the proposed system. Consider $P(X)=-X^2+\sum_1^na_jX^j$. This is a polynomial of degree smaller or equal to $n$, with $P(0)=0$. Moreover, by assumption $P(2^{i})=0$ for every $i=0,\ldots n-1$, so $P$ has at $n+1$ zeros, and it must be identically zero. So $(a_1,a_2,\ldots,a_n)=(0,1,\ldots,0)$. The converse is trivially true.

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  • $\begingroup$ Thanks for the answer. What do you mean by "must be identically zero"? $\endgroup$ – Sheheryar Zaidi Oct 4 '14 at 16:25
  • $\begingroup$ @SheheryarZaidi, A polynomial $P(x)$ of some degree $n$ that has distinct zeros in number exceeding $n$, must be zero for every $x$, that is $P(x)=0$ for every $x$. This is identically zero. Such a polynomial has all its coefficients equal to $0$. $\endgroup$ – Omran Kouba Oct 4 '14 at 16:35
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Use Cramer's rule remembering the $4^k=(2^k)^2$ which makes the right hand side the same as the second to last column. This means that two columns will be the same (and so the determinant will be zero) for all variables except for $a_2$ where the determinant in the numerator will be the same as the determinant in the denominator. Cramer's rule can be found here: http://en.wikipedia.org/wiki/Cramer's_rule

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