Exterior algebra is the algebra of alternating multilinear measurements. It explains determinants, oriented volume, cross products, and differential forms from a linear-algebraic point of view.
Alternating tensors
Definition. A \(k\)-linear form \(\omega:V^k\to F\) is alternating if it changes sign when two arguments are swapped, equivalently \(\omega(v_1, \ldots,v_k)=0\) whenever two arguments are equal. Alternation forces dependence to give zero volume.
Exterior powers
Definition. The \(k\)-th exterior power \(\bigwedge^kV\) is generated by symbols \[ v_1\wedge\cdots\wedge v_k \] that are multilinear and alternating in the vectors. The universal property says alternating \(k\)-linear maps \(V^k\to X\) correspond to linear maps \(\bigwedge^kV\to X\).
Theorem. If \(V\) has basis \((e_1, \ldots,e_n)\), then the wedge products \[ e_{i_1}\wedge\cdots\wedge e_{i_k}, \qquad 1\le i_1\lt\cdots\lt i_k\le n, \] form a basis of \(\bigwedge^kV\). Therefore \(\dim\bigwedge^kV=\binom nk\).
Proof. Multilinearity expands any wedge product into wedges of basis vectors. Alternation kills repeated indices and allows the remaining indices to be sorted at the cost of a sign. The sorted wedges span. Independence follows by constructing coordinate alternating forms that pick out one sorted multi-index and vanish on the others.
The determinant as top exterior power
Proposition. If \(\dim V=n\) and \(T:V\to V\), then \(\bigwedge^nT\) acts on the one-dimensional space \(\bigwedge^nV\) by multiplication by \(\det T\): \[ Tv_1\wedge\cdots\wedge Tv_n=(\det T)(v_1\wedge\cdots\wedge v_n). \]
Oriented volume
Definition. An orientation of a real \(n\)-dimensional vector space is a choice of which ordered bases are positive, where two bases have the same orientation if the change-of-basis determinant is positive. The element \(v_1\wedge\cdots\wedge v_n\) records oriented volume.
Example. In \(\mathbb R^3\), the cross product is a disguised exterior-algebra construction: \(u\times v\) corresponds to the two-vector \(u\wedge v\) after using the Euclidean inner product and orientation to identify \(\bigwedge^2\mathbb R^3\) with \(\mathbb R^3\).
Pullbacks of alternating forms
Proposition. If \(T:V\to W\) is linear and \(\omega\) is an alternating \(k\)-form on \(W\), the pullback \[ T^*\omega(v_1, \ldots,v_k)=\omega(Tv_1, \ldots,Tv_k) \] is an alternating \(k\)-form on \(V\). For top-degree forms, this pullback multiplies by the determinant when \(V=W\).
Remark. Exterior algebra is often where linear algebra becomes geometry. The wedge product refuses to count dependent directions twice, so it naturally measures area, volume, and orientation.
Area forms in coordinates
Calculation. In \(F^2\), the alternating form \(e^1\wedge e^2\) satisfies \[ (e^1\wedge e^2)((a,b),(c,d))=ad-bc. \] Thus the wedge product records the oriented area of the parallelogram spanned by the two coordinate vectors.
Example. In \(F^3\), \(u\wedge v\) represents the oriented plane element spanned by \(u\) and \(v\). Choosing an inner product and orientation lets one identify this bivector with the cross product \(u\times v\), but the bivector is the coordinate-free object.