structural:summarization, stylistic:style_transfer
Exterior power

The k th exterior power of V {\displaystyle V}, denoted ⋀ k (V) {\displaystyle {\textstyle \bigwedge }^{\!k}(V)}, is the vector subspace of ⋀ (V) {\displaystyle {\textstyle \bigwedge }(V)} spanned by elements of the form

If α ∈ ⋀ k (V) {\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}, then α {\displaystyle \alpha } is said to be a k -vector. If, furthermore, α {\displaystyle \alpha } can be expressed as an exterior product of k {\displaystyle k} elements of V {\displaystyle V}, then α {\displaystyle \alpha } is said to be decomposable (or simple, by some authors; or a blade, by others). Although decomposable k {\displaystyle k} -vectors span ⋀ k (V) {\displaystyle {\textstyle \bigwedge }^{\!k}(V)}, not every element of ⋀ k (V) {\displaystyle {\textstyle \bigwedge }^{\!k}(V)} is decomposable. For example, given R 4 {\displaystyle \mathbf {R} ^{4}} with a basis { e 1, e 2, e 3, e 4 } {\displaystyle \{e_{1},e_{2},e_{3},e_{4}\}}, the following 2-vector is not decomposable: