1
$\begingroup$
  1. I am going through a famous book on Error Control Coding (Channel Coding) to understand its basics. The author writes regarding Dual space as "Hence, Sd satisfies the two axioms for a subspace of a vector space over a finite field. Consequently, Sd is a subspace of the vector space Vn of all the n-tuples over GF(q). Sd is called the dual (or null ) space of S and vice versa."
  2. What i know from Linear Algebra is that a Dual space consists of set of all linear transformations on a vector space to the field F.
  3. At the same time there is another book that defines the concept in para 1 above with following wordings and with name of Dual SUBSPACE "If S is a k-dimensional subspace of the n-dimensional vector space Vn, the set Sd of vectors v for which for any u ∈ S and v ∈ Sd, u ◦ v = 0 is called the dual subspace of S"
  4. Null space is defined as all elements of the vector space which produce zero vector when a linear transformation is applied to them.

My questions are: a. Are the terminologies correct regarding the three concepts defined above (dual space, dual subspace and Null space). b. And the book referred in para 1 above have a typo?

$\endgroup$
0
$\begingroup$

In coding theory dual space is not used in the sense of linear functionals so forget about 2.

An $[n,k]$ linear code $C$ is a $k$ dimensional subspace of $\mathbb{F}_q^n$ (by definition) and its dual code $C^{\perp}$ is what is normally called the orthogonal complement of $C$ which is itself a subspace of dimension $n-k$.

$\endgroup$
1
  • $\begingroup$ Yes that is what i was confused about. Thanks $\endgroup$ – AJ HUNTER Sep 17 '20 at 18:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.