Wedge product/Elementary properties/Fact/Proof

(1) follows directly from construction.
(2). We consider the composed mapping

${\displaystyle V^{n}\longrightarrow H\cong K^{(V^{n})}\longrightarrow H/U,}$

where ${\displaystyle {}(v_{1},\ldots ,v_{n})}$ is sent to ${\displaystyle {}e_{(v_{1},\ldots ,v_{n})}}$, and this is sent to the residue class ${\displaystyle {}v_{1}\wedge \ldots \wedge v_{n}}$. The definition of the linear subspace ${\displaystyle {}U}$ ensures that the multilinearity and the alternating property are satisfied.
(3) hold, due to fact, for every alternating mapping.
(4). The first equation holds, due to fact, for every multilinear mapping. If in the index tuple ${\displaystyle {}(i_{1},\ldots ,i_{n})}$, an entry appears twice, then ${\displaystyle {}u_{i_{1}}\wedge \ldots \wedge u_{i_{n}}=0}$. Hence, we only have to consider tuples where all entries are different. After reordering, they have the form ${\displaystyle {}1\leq i_{1}<\ldots <i_{n}\leq m}$. For a fixed increasing index tuple, the sum over all permuted index tuples equals

${\displaystyle \sum _{\pi \in S_{n}}\prod _{j=1}^{n}a_{i_{\pi (j)}j}u_{i_{\pi (1)}}\wedge \ldots \wedge u_{i_{\pi (n)}}=\sum _{\pi \in S_{n}}\operatorname {sgn} (\pi )\prod _{j=1}^{n}a_{i_{\pi (j)}j}u_{i_{1}}\wedge \ldots \wedge u_{i_{n}}\,.}$