Consider the following definition from Loring W. Tu's An Introduction to Manifolds:
For a finite-dimensional vector space $V$, say of dimension $n$, define $$A_*(V)=\oplus_{k=0}^{\infty}A_k(V)=\oplus_{k=0}^{n}A_k(V)$$ where $A_0(V)={\mathbb R}$, and $A_k(k>0)$ denotes the set of all alternating $k$-linear functions $f$ on $V$, i.e., $$f:V^k\to{\mathbb R},\qquad f(v_{\sigma(1),\cdots,v_{\sigma(k)}})=(\text{sgn}\sigma)f(v_1,\cdots,v_k) \quad\text{for all} \quad\sigma\in S_k.$$ With the wedge product of multicovectors as multiplication, $A_*(V)$ becomes an anticommutative graded algebra, called the exterior algebra or the Grassmann algebra of multicovectors on the vector space $V$.
By definition of graded algebra, $A_*(V)$ has the structure of a vector space. But I don't understand what does the element of $A_*(V)$ look like. For example, if $f\in A_2(V)$ and $g\in A_3(V)$, then what is $f+g$? Since domains of $f$ and $g$ are of different dimensions, how can one "add" them?
So here are my questions:
What does the element of $A_*(V)$ look like? And what's the addition of the vector space?
According to the comments, the question above is actually a matter of understanding of the "direct sum". In stead of putting another post, I would like to ask a closed related question here:
How many different definitions are there of exterior algebra and how are they equivalent to each other?
I totally don't understand the one I learn from wikipedia. Any recommendation of a complete treatment of the subject?