Checking defining relation on a basis

Question

Assume I have two algebras $A,B$ over some field $k$ and a linear map $f:A\to B$. If I want to check that $f$ is a morphism of algebras, is it enough to check $f(ab)=f(a)f(b)$ on elements $a,b$ of a basis of $A$?

Answer 1

Let's prove your claim. Suppose $\left\{v_1, \dots, v_n\right\}$ is a basis of $A$. Now choose $x,y\in A$. Write $x=\sum_i\lambda_iv_i$ and $y=\sum_i\mu_iv_i$. Then \begin{eqnarray} f(xy) &=& \sum_{i,j}\lambda_i\mu_jf(v_iv_j)\\ &=& \sum_{i,j}\lambda_i\mu_jf(v_i)f(v_j)\\ &=& f(\sum_i\lambda_iv_i)f(\sum_j\mu_jv_j)\\ &=& f(x)f(y). \end{eqnarray} Examing this technique, it's clear that we only used that $\left\{v_1, \dots, v_n\right\}$ is generating (as a vector space). As mentioned in the comments, it also works for any generating set as an algebra by essentially the same proof.