# The exterior algebra

Similarly, the $${k^{\textrm{th}}}$$ exterior power of an n-dimensional vector space V  is defined to be

$$\displaystyle \Lambda^{k}V\equiv V\wedge V\wedge\ldots\wedge V\qquad(k\:\mathrm{times}).$$

The exterior product is generalized to $${\Lambda^{k}V}$$ by requiring the product to vanish if any two vector components are identical. This is equivalent to requiring the product to be completely anti-symmetric, i.e. to change sign under the exchange of any two vector components. Note that $${\Lambda^{k}V}$$ thus automatically vanishes for $${k>n}$$, since the $${(k+1)^{\textrm{th}}}$$ component will have to be a linear combination of previous components, resulting in terms $${v\wedge v=0}$$.

The exterior algebra (AKA Grassmann algebra, alternating algebra) is the tensor algebra modulo the relation $${v\wedge v\equiv0}$$, and can be written as

$$\displaystyle \Lambda V\equiv\Sigma\Lambda^{k}V=\mathbb{R\oplus}\Lambda^{1}V\oplus\Lambda^{2}V\oplus\dotsb\oplus\Lambda^{n}V,$$

where $${n}$$ is the dimension of $${V}$$ (since $${\Lambda^{k}V}$$ automatically vanishes for $${k>n}$$). The vector multiplication of the algebra is the exterior product $${\wedge}$$, which applied to $${A\in\Lambda^{j}V}$$ and $${B\in\Lambda^{k}V}$$ gives $${A\wedge B\in\Lambda^{j+k}V}$$ with the property

$$\displaystyle A\wedge B\equiv(-1)^{jk}B\wedge A.$$

If a pseudo inner product is defined on $${V}$$, it can be naturally extended to any $${\Lambda^{k}V}$$ by using the determinant: if $${A=v_{1}\wedge v_{2}\wedge\dotsb\wedge v_{k}}$$, and $${B=w_{1}\wedge w_{2}\wedge\dotsb\wedge w_{k}}$$, we define

$$\displaystyle \left\langle A,B\right\rangle \equiv\textrm{det}\left(\left\langle v_{i},w_{j}\right\rangle \right).$$

Note that this definition is also alternating, as it must for the inner product to be bilinear; i.e. exchanging two vectors in $${A}$$ reverses the sign of both $${A}$$ and $${\left\langle A,B\right\rangle }$$. Also note that if $${\hat{e}_{\mu}}$$ is an orthonormal basis for $${V}$$ and $${A=a\hat{e}_{1}\wedge\dotsb\wedge\hat{e}_{k}}$$, then $${\left\langle A,A\right\rangle =\pm a^{2}}$$. The pseudo inner product can then be extended to a nondegenerate symmetric multilinear alternating form on all of $${\Lambda V}$$ by defining it to be zero between elements from different exterior powers. In terms of the orthonormal $${\hat{e}_{\mu}}$$,

$$\displaystyle \left\{ \hat{e}_{\mu_{1}}\wedge\dotsb\wedge\hat{e}_{\mu_{k}}\right\} _{1\leq\mu_{1} < \dotsb < \mu_{k}\leq n}$$

is an orthonormal basis for $${\Lambda^{k}V}$$, and the union of such bases for $${k\leq n}$$ is an orthonormal basis for $${\Lambda V}$$.

In describing a particular element of $${\Lambda V}$$ we can unambiguously write $${+}$$ instead of $${\oplus}$$ and allow any zero terms to be omitted. A simple example can be helpful to keep in mind the concrete consequences of the exterior algebra’s abstract properties. The above depicts the exterior algebra over a real 2-dimensional vector space. Exterior products not shown above vanish; the product indicated with bold arrows is elaborated by the first equation. The second equation calculates the inner product of two elements of $${\Lambda^{2}V}$$ using the determinant.