How to check in the easiest way possible the linear independence of these code words?
They are supposed to be linear independent, but I cannot find out why...
1st) 10101 00101 100
2nd) 00111 01100 101
3rd) 01001 10101 111
4th) 10011 01011 101
5th) 01100 11100 100
I've been told that, just by looking and the ones on each of them, I can discover quickly that they are linear independent, but how?
NB: There are similar questions but none of them address the down-to-earth method doable by a human to check linear independence, so, as far as I know, this is not duplicated
So, suppose you had a dependence, $$\sum_{i=1}^5a_i\vec {v_i}$$
We want to prove that all the $a_i$ are $0$.
Looking at the second to last column, we see that $a_3=0$.
Looking at the sixth column shows that $$a_3+a_5=0\implies a_5=0$$
Looking at the first column we see that $$a_1+a_4=0$$
Looking at the last column we see that $$a_2+a_4=0\implies a_2=a_1=-a_4$$
Looking at the third column tells us that $$a_1+a_2+a_5=0\implies a_1+a_2=0\implies a_1=a_2=-a_4=0$$ and we are done.