Decomposable $k$-vectors and the basic wedge product

For $v_1,\dots ,v_k\in V$, the image of the basic wedge map of The $k$th exterior power as the tensor-power quotient by repeated-vector relations is written

$v_1\wedge \cdots \wedge v_k:=v_1\otimes \cdots \otimes v_k+W_k\in {\mathrm{\Lambda }}^{k}V,$

and is called the wedge (or exterior product) of the list. A $k$-vector $\alpha \in {\mathrm{\Lambda }}^{k}V$ is decomposable (or pure) when it equals $v_1\wedge \cdots \wedge v_k$ for some list; a general element of ${\mathrm{\Lambda }}^{k}V$ is a finite sum of decomposables, because the pure tensors span $V^{\otimes k}$ and their classes span the quotient.

For $k=0$ the empty wedge is $1\in F={\mathrm{\Lambda }}^{0}V$; for $k=1$ the wedge of a single vector is the vector itself under the identification ${\mathrm{\Lambda }}^{1}V=V$.

The wedge of $v_1,\dots ,v_k$ depends on the order of the list only up to a sign (see Exterior multiplication is well defined, graded, associative, unital, and graded-commutative); the notation $v_1\wedge \cdots \wedge v_k$ records the order.