## Tuesday, October 30, 2007

### Linear Independence

This problem is the generalization of exercise 2.3.9 from 'linear algebra' Hoffman and Kunze.

Let the set of vectors ${a_1,a_2,...,a_n}$ be linearly independent. Prove that ${v_1,v_2,...,v_n}$ are also linearly independent given

$v_i=\sum_{cyclic} {k_i + k_{i+1} +...+k_{(i+n-1)mod n}$

I mean
$v_1=a_1 + a_2 +...+ a_{n-1}$
$v_2=a_1 + a_2 +...+ a_n$
and so on.

ps: I proved it by proving a certain matrix to be invertible and that was pretty cumbersome. Hoping for a better pretty solution :)