Let K {\displaystyle {}K} be a field, let V {\displaystyle {}V} and W {\displaystyle {}W} be vector spaces over K {\displaystyle {}K} , and n ∈ N {\displaystyle {}n\in \mathbb {N} } . Show that the set of all alternating mappings (denoted by Alt K n ( V , W ) {\displaystyle {}\operatorname {Alt} _{K}^{n}{\left(V,W\right)}} ) is a linear subspace of Mult K ( V , … , V , W ) {\displaystyle {}\operatorname {Mult} _{K}{\left(V,\ldots ,V,W\right)}} (where the vector space V {\displaystyle {}V} appears n {\displaystyle {}n} -fold).
be a
field, let 
and 
be
vector spaces
over , and
.
Show that the set of all
alternating
mappings
(denoted by 
)
is a
linear subspace
of 
(where the vector space appears 
-fold).
