According to 'an introduction to manifolds' written by Tu, he defined the exterior algebra as \begin{equation} A_*(V)=\oplus^\infty_{k=0}A_k(V) \end{equation} where $V$ is a vectorspace with dimension $n$ and $A_k(V)$ is a set of alternating $k$tensors.
here I have a question. The book said that multiplication in this algebra is the wedge product.
But if $a^i\in A_{k_i}(V),b^j\in A_{l_j}(V)$ then \begin{equation} a=(a^1,... a^{k'})\oplus^\infty_{k=0}A_k(V),(b^1,... b^l)\in\oplus^\infty_{k=0}A_k(V) \end{equation} Then I don't know how to multiply these $a$ and $b$ through wedge product.