Editing Exterior algebra (section)

== Motivating examples ==
<!--The purpose of this section is to motivate the skewness of the exterior product on vectors in ''V''-->

=== Areas in the plane ===
[[Image:Area parallellogram as determinant.svg|thumb|right|The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices.]]

The two-dimensional [[Euclidean vector space]] <math> \mathbf{R}^2 </math> is a [[Real number|real]] vector space equipped with a [[basis of a vector space|basis]] consisting of a pair of orthogonal [[unit vector]]s
<math display=block>\mathbf{e}_1 = \begin{bmatrix}1\\0\end{bmatrix},\quad \mathbf{e}_2 = \begin{bmatrix}0\\1\end{bmatrix}.</math>

Suppose that
<math display=block>\mathbf{v} = \begin{bmatrix}a\\b\end{bmatrix}
= a \mathbf{e}_1 + b \mathbf{e}_2, \quad \mathbf{w}
= \begin{bmatrix}c\\d\end{bmatrix}
= c \mathbf{e}_1 + d \mathbf{e}_2
</math>
are a pair of given vectors in {{tmath|\mathbf{R}^2}}, written in components.  There is a unique parallelogram having <math>\mathbf{v}</math> and <math>\mathbf{w}</math> as two of its sides.  The ''area'' of this parallelogram is given by the standard [[determinant]] formula:
<math display=block>
\text{Area}
= \left| \det \begin{bmatrix} \mathbf{v} & \mathbf{w} \end{bmatrix} \right|
= \left| \det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \right|
= \left| ad - bc \right| .
</math>

Consider now the exterior product of <math>\mathbf{v}</math> and {{tmath|\mathbf{w} }}:
<math display=block>\begin{align}
\mathbf{v} \wedge \mathbf{w}
&= (a\mathbf{e}_1 + b\mathbf{e}_2) \wedge (c\mathbf{e}_1 + d\mathbf{e}_2) \\
&= ac\mathbf{e}_1 \wedge \mathbf{e}_1 + ad\mathbf{e}_1 \wedge \mathbf{e}_2 + bc\mathbf{e}_2 \wedge \mathbf{e}_1 + bd\mathbf{e}_2 \wedge \mathbf{e}_2 \\
&= \left( ad - bc \right)\mathbf{e}_1 \wedge \mathbf{e}_2,
\end{align}</math>
where the first step uses the distributive law for the [[#Formal definition|exterior product]], and the last uses the fact that the exterior product is an [[alternating map]], and in particular <math>\mathbf{e}_2 \wedge \mathbf{e}_1 = -(\mathbf{e}_1 \wedge \mathbf{e}_2).</math>  (The fact that the exterior product is an alternating map also forces <math> \mathbf{e}_1 \wedge \mathbf{e}_1 = \mathbf{e}_2 \wedge \mathbf{e}_2 = 0.</math>) Note that the coefficient in this last expression is precisely the determinant of the matrix {{nowrap|['''v''' '''w''']}}.  The fact that this may be positive or negative has the intuitive meaning that '''v''' and '''w''' may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define.  Such an area is called the [[signed area]] of the parallelogram: the [[absolute value]] of the signed area is the ordinary area, and the sign determines its orientation.

The fact that this coefficient is the signed area is not an accident.  In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct.  In detail, if {{nowrap|1=A('''v''', '''w''')}} denotes the signed area of the parallelogram of which the pair of vectors '''v''' and '''w''' form two adjacent sides, then A must satisfy the following properties:
# {{nowrap|1=A(''r'''''v''', ''s'''''w''') = ''rs''A('''v''', '''w''')}} for any real numbers ''r'' and ''s'', since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram).
# {{nowrap|1=A('''v''', '''v''') = 0}}, since the area of the [[degenerate (mathematics)|degenerate]] parallelogram determined by '''v''' (i.e., a [[line segment]]) is zero.
# {{nowrap|1=A('''w''', '''v''') = −A('''v''', '''w''')}}, since interchanging the roles of '''v''' and '''w''' reverses the orientation of the parallelogram.
# {{nowrap|1=A('''v''' + ''r'''''w''', '''w''') = A('''v''', '''w''')}} for any real number ''r'', since adding a multiple of '''w''' to '''v''' affects neither the base nor the height of the parallelogram and consequently preserves its area.
# {{nowrap|1=A('''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = 1}}, since the area of the unit square is one.
[[Image:Exterior calc cross product.svg|upright=1.2|thumb|The cross product ('''<span style="color:blue;">blue</span>''' vector) in relation to the exterior product ('''<span style="color:#779ECB;">light blue</span>''' parallelogram).  The length of the cross product is to the length of the parallel unit vector ('''<span style="color:#CC0000;">red</span>''') as the size of the exterior product is to the size of the reference parallelogram ('''<span style="color:#CC4E5C;">light red</span>''').]]
With the exception of the last property, the exterior product of two vectors satisfies the same properties as the area.  In a certain sense, the exterior product generalizes the final property by allowing the area of a parallelogram to be compared to that of any chosen parallelogram in a parallel plane (here, the one with sides '''e'''<sub>1</sub> and '''e'''<sub>2</sub>).  In other words, the exterior product provides a ''basis-independent'' formulation of area.<ref>This axiomatization of areas is due to [[Leopold Kronecker]] and [[Karl Weierstrass]]; see {{harvtxt|Bourbaki|1989b|loc=Historical Note}}.  For a modern treatment, see {{harvtxt|Mac Lane|Birkhoff|1999|loc=Theorem IX.2.2}}.  For an elementary treatment, see {{harvtxt|Strang|1993|loc=Chapter 5}}.</ref>

===Cross and triple products===
[[File:2-vector decomposition.png|thumb|right|Basis Decomposition of a 2-vector]]
For vectors in '''R'''<sup>3</sup>, the exterior algebra is closely related to the [[cross product]] and [[triple product]]. Using the standard basis {{nowrap|{{mset|'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>}}}}, the exterior product of a pair of vectors
: <math> \mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3 </math>
and
: <math> \mathbf{v} = v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2 + v_3 \mathbf{e}_3 </math>
is
: <math> \mathbf{u} \wedge \mathbf{v} = (u_1 v_2 - u_2 v_1) (\mathbf{e}_1 \wedge \mathbf{e}_2) </math>
: <math> \phantom{\mathbf{u} \wedge \mathbf{v}} + (u_3 v_1 - u_1 v_3) (\mathbf{e}_3 \wedge \mathbf{e}_1) </math>
: <math> \phantom{\mathbf{u} \wedge \mathbf{v}} + (u_2 v_3 - u_3 v_2) (\mathbf{e}_2 \wedge \mathbf{e}_3) </math>
where {{mset|'''e'''<sub>1</sub> ∧ '''e'''<sub>2</sub>, '''e'''<sub>3</sub> ∧ '''e'''<sub>1</sub>, '''e'''<sub>2</sub> ∧ '''e'''<sub>3</sub>}} is the basis for the three-dimensional space ⋀<sup>2</sup>('''R'''<sup>3</sup>). The coefficients above are the same as those in the usual definition of the [[cross product]] of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a [[bivector]].

Bringing in a third vector
: <math> \mathbf{w} = w_1 \mathbf{e}_1 + w_2 \mathbf{e}_2 + w_3 \mathbf{e}_3, </math>
the exterior product of three vectors is
: <math> \mathbf{u} \wedge \mathbf{v} \wedge \mathbf{w} = (u_1 v_2 w_3 + u_2 v_3 w_1 + u_3 v_1 w_2 - u_1 v_3 w_2 - u_2 v_1 w_3 - u_3 v_2 w_1) (\mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3) </math>
where '''e'''<sub>1</sub> ∧ '''e'''<sub>2</sub> ∧ '''e'''<sub>3</sub> is the basis vector for the one-dimensional space ⋀<sup>3</sup>('''R'''<sup>3</sup>). The scalar coefficient is the [[triple product]] of the three vectors.

The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product {{nowrap|'''u''' × '''v'''}} can be interpreted as a vector which is perpendicular to both '''u''' and '''v''' and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the [[minor (mathematics)|minors]] of the matrix with columns '''u''' and '''v'''. The triple product of '''u''', '''v''', and '''w''' is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns '''u''', '''v''', and '''w'''. The exterior product in three dimensions allows for similar interpretations. In fact, in the presence of a positively oriented [[orthonormal basis]], the exterior product generalizes these notions to higher dimensions.