Here is the question I am trying to solve:
(Divided powers) Consider the vector space $C = k[t]$ of polynomials in one variable. Prove that there exists a unique coalgebra structure $(C, \Delta, \varepsilon)$ on $C$ such that $$\Delta(t^n) = \sum_{p + q = n} t^p \otimes t^q \text{ and } \varepsilon(t^n) = \delta_{n0}$$ for all $n \geq 0.$ Show that $C$ becomes a bialgebra when given the product $$t^pt^q = t^p t^q = {p+q \choose p} t^{p+q}.$$
My questions are:
1-To prove the first part of the question the two maps $\Delta, \epsilon$ should be linear but I do not know the reason they are linear. Could someone clarify this to me please?
2- To prove the second part of the question, the two maps $\eta, \mu $ should be linear but I do not know the reason, could someone clarify this to me please? $\eta$ (defined on the field $k$) and $\mu$(defined on a tensor product of basis elements)
Here are the figures describing the commutativity of the 4 maps:
