# “extending linearly”

I defined the product $m:H \otimes H \rightarrow H$ in the group ring $H={\mathbb{K}}G$ by “setting $m(g \otimes h) = gh$ and extending linearly”. A couple of people asked me what I meant by that.

I am referring to the fact that a linear function is determined uniquely by its values on a basis. More precisely, suppose I want to define a linear function $f:V \rightarrow W$ between vector spaces, and I have a basis $\{v_i\}_{i\in I}$ for $V$. Then:

a) Once you choose $f(v_i)$ for each $i \in I$, you will have determined $f(v)$ for all $v \in V$, and

b) You may choose $f(v_i)$ for each $i \in I$ in any way that you want (as long as they are in $W$).

In the situation above, the elements $g \otimes h$ (for $g, h \in G$) form a basis for ${\mathbb{K}}G$. What I said simply means that I am defining $m(\sum_i \alpha_i (g_i \otimes h_i)) = \sum_i \alpha_i(g_ih_i)$ for $\alpha_i \in \mathbb{K}, g_i, h_i \in G$ (which is more explicit but messier when written like this).