# Difference between Dual space, dual subspace of a Vector Space.

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?

## 1 Answer

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 $$n-k$$.

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