Consider a graded vector space $V$ with basis $\{a,b\}$ such that $a\in {V}^{2}$ and $b\in {V}^{5}$. Now define a linear map $d$ (of degree 1) by $da=0$ and $db={a}^{3}$. It follows that d extends uniquely to a derivation $d:\mathrm{\Lambda}V\to \mathrm{\Lambda}V$.

The point of the example is to show that the derivation on $\mathrm{\Lambda}V$ is completely determined by its values on $V$. So if i understand well, he considers a linear map $d:V\u27f6\mathrm{\Lambda}V$ of degree one defined by

(here ${\mathrm{\Lambda}}^{k}V$ is the set of elements of word length $k$) and

${d}_{5}:{V}^{5}\u27f6{\mathrm{\Lambda}}^{6}V;b\mapsto {a}^{3}$

The first question that i'm stuck on is for ${d}_{2}(b)={a}^{3}$, i mean ${a}^{3}$ is of length $3$, how it can be in ${\mathrm{\Lambda}}^{6}V$.