Minimum count of linear dependent columns in tensor product - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:50:27Z http://mathoverflow.net/feeds/question/61898 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61898/minimum-count-of-linear-dependent-columns-in-tensor-product Minimum count of linear dependent columns in tensor product spk 2011-04-16T08:08:56Z 2011-04-19T15:02:44Z <p>Good day!</p> <p>I have a Vandermonde-like matrix $V = \left( \begin{array}{ccccc} 1 &amp; \alpha &amp; \alpha^2 &amp; \ldots &amp; \alpha^{n-1} \newline 1 &amp; \alpha^2 &amp; \alpha^4 &amp; \ldots &amp; \alpha^{2n - 2} \newline &amp; &amp; \vdots &amp; &amp; \newline 1 &amp; \alpha^{n-1} &amp; \alpha^{2n-2} &amp; \ldots &amp; \alpha^{(n-1)^2} \end{array} \right)$, where $\alpha$ is $n$-th root from unity in some extended field $GF(2^r)$. So sum of all columns of $V$ vanishes. And no other columns (of $V$) combinations vanishes over $GF(2)$.</p> <p>After that I consider tensor product of $V$ with row $\mathbf{a} = (a_1, a_2, \ldots, a_{l-1})$ over the same field $GF(2^r)$. I know that elements of $\mathbf{a}$ are linear independent over $GF(2)$. </p> <p>Now I consider tensor product $P = V \otimes \mathbf{a}$. Is it right, sum of no $p &lt; n$ columns of $P$ vanishes over $GF(2)$?</p> <p>Thank you!</p>