What is this vector space: $∧^k(V)$ (where $V$ is some vector space over an arbitrary field)? I know that it is spanned by vectors or objects of the form $v_1∧v_2∧...∧v_k$ where each $v_i \in V$.  But I am unsure of what this operation is or what such a vector or object would "look" like. 
I need to research this vector space because my notes are inadequate.  I was falling asleep during lecture and two cans of redbull just made me fly to the bathroom... nothing else... Can somebody give me a basic explanation of what this space is and behaves like?  Or point me in the right direction with some links?    
 A: Are you familiar with tensor products? This is essentially a tensor product, with the added condition that the tensoring is "anti-commutative". That is, for all $v,w\in V$, we have $v\wedge w=-(w\wedge v)$. More generally, this implies that if $v_1,\dots,v_k\in V$ and $\sigma\in S_k$ (the permutation group on $k$ elements), then
$$v_{\sigma(1)}\wedge\cdots\wedge v_{\sigma(k)}=\operatorname{sgn}(\sigma)(v_1\wedge\cdots\wedge v_k).$$
To define it formally, we start with the tensor algebra on $V$:
$$T(V)=\oplus_{k\ge0}V^{\otimes k}.$$
Then we define $\Lambda(V)=T(V)/I$ where $I$ is the ideal generated by elements of the form $v\otimes w+w\otimes v$ for $v,w\in V$. Then we write $v_1\wedge\cdots\wedge v_k$ for the representative of $v_1\otimes\cdots\otimes v_k$, and define $\Lambda^k(V)$ to be the subspace spanned by elements of this form.
A: This is called the $k^{th}$ exterior power of the vector space $V$. It is a summand of the so-called Grassmann (or Exterior) Algebra of $V$,
$$ \Lambda(V)=\bigoplus_{k\in \mathbf{N}}\Lambda^k(V).$$
The notes found here may help you:
$(1)$ 
http://www.math.uiuc.edu/~lerman/519/s11/mult.pdf
$(2)$ 
https://people.maths.ox.ac.uk/hitchin/hitchinnotes/Projective_geometry/Chapter_3_Exterior.pdf
