What does it mean for a subspace (of a finite dimensional vector space over a field) to be linear?

Question

In my lecture notes on coding theory (which is heavily based on linear algebra), it used the following definition:

Suppose that $F$ is a field, and denote by $F^n$ the $n$-dimensional vector space over $F$. We say that a code $C \subset F^n$ is a linear code if $C$ is a linear subspace of $F^n$.

I understand what it means for $C$ to be a subspace of the field $F^n$, but what does it mean for a subspace of $F^n$ to be linear?

Answer 1

It means the same thing as it does in the special case of $\Bbb F = \Bbb R$. The subset $C\subseteq \Bbb F^n$ is linear iff all the following are satisfied:

1. $(0,0,\ldots,0)\in C$
2. If $x, y\in C$ then $x+y\in C$ (where $+$ denotes the standard vector addition)
3. If $x\in C$ and $k\in \Bbb F$, then $k\cdot x\in C$ (where $\cdot$ denotes the standard scalar vector multiplication)

Answer 2

If you understand subspace as something like "hyperplane through the origin" or "set of vectors closed under addition and scalar multiplication" then "linear" adds no extra information. It just means we want that property as opposed to something else that someone might call "subspace" like maybe "subset that is a topological space with inherited topology".