Posted by: atri | February 4, 2011

Lecture 7: More on linear codes

In today’s lecture we looked at the notion of the null/dual space of a linear subspace and then saw some consequences. In particular, we saw that the existence of a parity check matrix implies polynomial time error detection for linear codes. I typed up some notes for the stuff on null spaces. The notes have a slightly better organization wrt the definition of a dual space and its consequences. The implications for linear codes are from Lecture 5 from Fall 2007.

Also I should correct a brain-freeze I had in the lecture today. While talking about linear subspaces, Jeff noted that a linear subspace and its dual can have the same vector in its basis. Then Dan observed that this is not true for linear subspace over the reals. I kind of hedged it in class but Dan was correct. Even though I don’t quite like the reals, I should not have missed the very simple argument behind Dan’s claim– sorry about that! Here is a quick proof: let $\mathbf{v}=(v_1,\dots,v_n)\in\mathbb{R}^n$. Then note that $\langle \mathbf{v},\mathbf{v}\rangle=\sum_{i=1}^n v_i^2\ge 0$, where the equality only holds when $\mathbf{v}=\mathbf{0}$. In other words, $\mathbf{v}\in S\cap S^{\perp}$  (for a linear subspace $S\subseteq \mathbb{R}^n$) if and only if $\mathbf{v}=\mathbf{0}$, which of course means that $\mathbf{v}$ cannot be an element of a basis.