“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).

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