Exterior power of a vector space

For a vector space $V$ over $K$ and $k\ge1$ , its $k$ -th exterior power is a vector space $\Lambda^k V$ equipped with an alternating multilinear map

\iota:V^k\to\Lambda^kV,\qquad (v_1,\ldots,v_k)\mapsto v_1\wedge\cdots\wedge v_k,

satisfying the universal property: for every vector space $W$ and alternating multilinear map $f:V^k\to W$ , there exists a unique linear map $\widetilde f:\Lambda^kV\to W$ with $f=\widetilde f\circ\iota$ . Set $\Lambda^0V=K$ .

Basis and low degrees

If $e_1,\ldots,e_n$ is a basis of $V$ , the wedges $e_{i_1}\wedge\cdots\wedge e_{i_k}$ with $i_1<\cdots<i_k$ form a basis of $\Lambda^kV$ . Thus $\dim\Lambda^kV=\binom nk$ , and $\Lambda^kV=0$ for $k>n$ . There is a canonical identification $\Lambda^1V=V$ .

Relation to the exterior algebra

These spaces are the graded pieces of the exterior algebra of $V$ . The universal property determines each exterior power up to the unique isomorphism preserving its distinguished alternating map.

References

Bernhard Leeb, Some multilinear algebra, 2020, §§1.1, 1.4 and 2.3.