- 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."
- 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.
- 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"
- 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?
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 $n-k$.