# Understanding Exterior Algebra: Determinant from Wedge Product

• Matthollyw00d
In summary, the wedge product is a mathematical operation that is used in exterior algebra and differential geometry. It is defined as the product of two vectors that results in a new object called a "bivector." This product is required to change sign on alternation and is linear in either term. The wedge product also has the property that a ^ a = 0 for any simple element a. It is often used in the calculation of determinants, where it appears as a factor. Some recommended books on this topic are Geometric Algebra for Computer Science and New Foundations of Classical Mechanics.
Matthollyw00d
Can someone please thoroughly explain how the determinant comes from the wedge product? I'm only in Cal 3 and Linear at the moment. I'm somewhat trying to learn more about the Wedge Product in Exterior Algebra to understand the determinant on a more fundamental basis. A thorough website or book would be of great help also.

The wedge product required for this is that it changes sign on alternation, and is linear in either term

ie:
$$b \wedge a = - a \wedge b$$

and
$$(a + b) \wedge c = a \wedge c + b \wedge c$$

and

$$a \wedge (b+c) = a \wedge b + a \wedge c$$(the first assumes one is talking about what is referred to as a simple element, one that can be built up of wedge products of other simple elements).

Note that a consequence of the first is that a ^ a = 0 for any simple element a (for example a vector in the two element case).

As an illustration of how a determinant enters the picture consider the two variable case. The wedge products of two vectors in a plane

\begin{align*} (a_1 e_1 + a_2 e_2 ) \wedge (b_1 e_1 + b_2 e_2 ) &= a_1 b_1 e_1 \wedge e_1 +a_1 b_2 e_1 \wedge e_2 +a_2 b_1 e_2 \wedge e_1 +a_2 b_2 e_2 \wedge e_2 \\ &= a_1 b_2 e_1 \wedge e_2 +a_2 b_1 e_2 \wedge e_1 \\ &= a_1 b_2 e_1 \wedge e_2 -a_2 b_1 e_1 \wedge e_2 \\ &= (a_1 b_2 -a_2 b_1) e_1 \wedge e_2 \end{align*}

Observe the determinant above as a factor of the wedge product.

edit. As for books I'd personally recommend Geometric Algebra for Computer Science, and New Foundations of Classical Mechanics (but be warned that neither of these exclusively treat the wedge product and exterior algebra nor are about differential forms if that is what you are looking for).

Last edited:
Peeter said:
The wedge product required for this is that it changes sign on alternation, and is linear in either term

ie:
$$b \wedge a = - a \wedge b$$

and
$$(a + b) \wedge c = a \wedge c + b \wedge c$$

and

$$a \wedge (b+c) = a \wedge b + a \wedge c$$

(the first assumes one is talking about what is referred to as a simple element, one that can be built up of wedge products of other simple elements).

Note that a consequence of the first is that a ^ a = 0 for any simple element a (for example a vector in the two element case).

As an illustration of how a determinant enters the picture consider the two variable case. The wedge products of two vectors in a plane

\begin{align*} (a_1 e_1 + a_2 e_2 ) \wedge (b_1 e_1 + b_2 e_2 ) &= a_1 b_1 e_1 \wedge e_1 +a_1 b_2 e_1 \wedge e_2 +a_2 b_1 e_2 \wedge e_1 +a_2 b_2 e_2 \wedge e_2 \\ &= a_1 b_2 e_1 \wedge e_2 +a_2 b_1 e_2 \wedge e_1 \\ &= a_1 b_2 e_1 \wedge e_2 -a_2 b_1 e_1 \wedge e_2 \\ &= (a_1 b_2 -a_2 b_1) e_1 \wedge e_2 \end{align*}

Observe the determinant above as a factor of the wedge product.

edit. As for books I'd personally recommend Geometric Algebra for Computer Science, and New Foundations of Classical Mechanics (but be warned that neither of these exclusively treat the wedge product and exterior algebra nor are about differential forms if that is what you are looking for).

I'm also somewhat confused on notation. The 2 vectors you're using in your example are A= [a1 a2] and B= [b1 b2], correct? Then are the e1 and e2, unit vectors, like i, j, and k? And then what does that final e1 wedge e2 do to vanish when dealing with determinants?

e1 and e2 are two vectors that aren't colinear, but they can be i and j if you like. If you use a = a_1 i + a_2 j + a_3 k, and b = b_1 i + ...
then you'll get something like:

$$a \wedge b = \begin{vmatrix} a_1 & a_2 \\ b_1 & b_2 \end{vmatrix} i \wedge j + \begin{vmatrix} a_2 & a_3 \\ b_2 & b_3 \end{vmatrix} j \wedge k + \begin{vmatrix} a_1 & a_3 \\ b_1 & b_3 \end{vmatrix} i \wedge k$$

The final wedges do not vanish unless you'll chosen i, j, k to be linearly dependent (which wouldn't be the case if this is your standard orthonormal basis for R^3). The wedge products i^j, j^k, i^k can be thought of as forming a basis in a "wedge product" space in their own right.

Last edited:

## 1. What is exterior algebra?

Exterior algebra is a mathematical structure that extends the concepts of vectors and matrices to higher dimensions by introducing new operations such as the wedge product and the exterior derivative.

## 2. What is the determinant in exterior algebra?

In exterior algebra, the determinant is a function that maps a square matrix to a scalar value, representing the volume of the parallelepiped spanned by the column vectors of the matrix. It is calculated using the wedge product of the column vectors.

## 3. What is the wedge product in exterior algebra?

The wedge product is a binary operation in exterior algebra that takes two vectors and produces a new vector called the exterior product. It is used to calculate the determinant and other important concepts in exterior algebra.

## 4. How is exterior algebra related to linear algebra?

Exterior algebra is a generalization of linear algebra, as it extends the concept of vector spaces to include non-commutative operations such as the wedge product. It also provides a geometric interpretation of linear algebra concepts, such as determinants and cross products.

## 5. Why is understanding exterior algebra important?

Exterior algebra is a powerful mathematical tool that has applications in various fields, including physics, engineering, and computer science. It allows for a deeper understanding of geometric concepts and provides a more elegant approach to solving problems involving vectors and matrices in higher dimensions.

• Linear and Abstract Algebra
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
8
Views
3K
• Linear and Abstract Algebra
Replies
19
Views
697
• Differential Geometry
Replies
13
Views
893
• Linear and Abstract Algebra
Replies
32
Views
4K
• Linear and Abstract Algebra
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
1
Views
2K
• Linear and Abstract Algebra
Replies
1
Views
940
• Linear and Abstract Algebra
Replies
6
Views
5K
• Linear and Abstract Algebra
Replies
5
Views
2K