Question

Show that in Grassmann's combinatory multiplication, $$ \begin{aligned} &\left(\sum \alpha_{1 i} \epsilon_{i}\right)\left(\sum \alpha_{2 i} \epsilon_{i}\right) \cdots\left(\sum \alpha_{n i} \epsilon_{i}\right) \\ &\quad=\operatorname{det}\left(\alpha_{i j}\right)\left[\epsilon_{1} \epsilon_{2} \cdots \epsilon_{n}\right] \end{aligned} $$ where each linear combination is of a given set of \(n\) firstorder units and where \(\left[\epsilon_{1} \epsilon_{2} \cdots \epsilon_{n}\right]\) is the single unit of \(n\)th order.

Short answer

Question: Show that Grassmann's algebraic expression for a product of linear combinations of first-order units can be rewritten as the determinant of a matrix times a single unit of nth order. Answer: By expanding the expression and using the antisymmetric properties of basis vectors, we can simplify the product of linear combinations of first-order units to a sum of unique permutations of indices. This sum can be identified as the product of the determinant of a matrix times a single unit of the nth order, as shown in the step-by-step solution above: \(\operatorname{det}(\alpha_{ij}) [\epsilon_{1} \epsilon_{2} \cdots \epsilon_{n}]\)

Step-by-step solution

Identify the components of the expression

We need to understand the given expression before moving forward with the solution. The given expression is a product of linear combinations of first-order units, which can be written as: \((\sum_{i} \alpha_{1i} \epsilon_{i})(\sum_{i} \alpha_{2i} \epsilon_{i}) \cdots(\sum_{i} \alpha_{ni} \epsilon_{i})\) We are also given the single unit of nth order as: \([\epsilon_{1} \epsilon_{2} \cdots \epsilon_{n}]\)

Expanding the expression

To work with this expression, we first need to expand it by multiplying each term. As each term in every linear combination is composed of a scalar (\(\alpha_{ij}\)) and a basis vector (\(\epsilon_{i}\)), a multiplication of two terms will result in a scalar and some product of two basis vectors. For example: \((\alpha_{1i} \epsilon_{i})(\alpha_{2j} \epsilon_{j}) = \alpha_{1i} \alpha_{2j} (\epsilon_{i} \epsilon_{j})\) After this multiplication, the expression becomes the sum over all products of these terms, the scalar coefficients, and the cross products of basis vectors. Summing over all terms in the expanded expression, we get: \(\sum_{i_{1}, i_{2}, ... ,i_{n}} \alpha_{1 i_{1}} \alpha_{2 i_{2}} \cdots \alpha_{n i_{n}} (\epsilon_{i_{1}} \epsilon_{i_{2}} \cdots \epsilon_{i_{n}})\) Notice that we are now working with an expression that involves the cross products of basis vectors.

Identify Antisymmetric properties of basis vectors

The product of basis vectors has an antisymmetric property. It means that if two basis vectors are the same, their product equals 0. In other words: \(\epsilon_{i} \epsilon_{j} = -\epsilon_{j} \epsilon_{i}\) for \(i \neq j\) and \(\epsilon_{i} \epsilon_{i} = 0\) for all \(i\) This property allows us to simplify our expression further.

Simplify the expression using antisymmetric properties

By using the antisymmetric properties of basis vectors, we can remove all terms in our expression where two or more indices are the same. The only nonzero terms are products where all indices are different. So, the expression simplifies to a sum of unique permutations of indices: \(\sum_{\sigma} \alpha_{1 \sigma(1)} \alpha_{2 \sigma(2)} \cdots \alpha_{n \sigma(n)} (\epsilon_{\sigma(1)} \epsilon_{\sigma(2)} \cdots \epsilon_{\sigma(n)})\) Where the summation is over all unique permutations of indices \(\sigma\).

Identify the determinant and the product of basis vectors

At this point, we can notice that the sum is equivalent to the product of the determinant of the matrix \(\alpha_{ij}\) and the product of basis vectors: \(\operatorname{det}(\alpha_{ij}) [\epsilon_{1} \epsilon_{2} \cdots \epsilon_{n}]\) Remember that the determinant of a square matrix is the sum of the product of elements in any row or column multiplied by their corresponding signed minors. This property allows us to identify the determinant in the sum in our expression: \(\operatorname{det}(\alpha_{ij}) = \sum_{\sigma} \text{sgn}(\sigma) \alpha_{1 \sigma(1)} \alpha_{2 \sigma(2)} \cdots \alpha_{n \sigma(n)}\) So the expression simplifies to: \(\operatorname{det}(\alpha_{ij}) [\epsilon_{1} \epsilon_{2} \cdots \epsilon_{n}]\) Therefore, we have shown that Grassmann's combinatory multiplication can be expressed as the determinant of a matrix times a single unit of nth order.

Key concepts

Mathematical Proof

In mathematics, a proof is a logical argument demonstrating that a specific statement, theorem, or formula is indisputably true. A mathematical proof must establish a claim using deductive reasoning and must adhere to the rules of logic and evidence. The essence of demonstrating Grassmann's combinatory multiplication relies on these principles of proofs.

In the context of Grassmann's multiplication, we are dealing with a complex algebraic expression that involves summing products of vectors and scalars. The aim is to prove that this summation equates to the determinant of a matrix multiplied by a specific combination of unit vectors. Students can greatly benefit from a step-by-step approach that methodically expands, simplifies, and relates the expression to known mathematical constructs like determinants and basis vectors with antisymmetric properties. Exploring each step in detail using logical rules provides not just the understanding of this particular proof, but also a strong example of how mathematical proofs are constructed in general.

Determinants

Determinants are a fundamental concept in linear algebra, associated with square matrices. The determinant of a matrix is a single number that provides useful information about the matrix, such as whether the matrix is invertible, and the volume of the geometric transformation defined by the matrix.

The standard notation for the determinant of a matrix 'A' is \(\det(A)\) or |A|. It is calculated via multiple methods, including expansion by minors, which involves removing row and column elements. A key property of the determinant is its ability to be expressed as a sum of permutations of its elements, each multiplied by the sign of the permutation \(\text{sgn}(\sigma)\).

By decomposing the problem of calculating determinants into elementary row operations, matrix cofactors, and the understanding of permutation signs, students can grasp why the alternation of signs is necessary and how it reflects the antisymmetric nature of the determinant.

Linear Algebra

Linear algebra is a branch of mathematics concerning linear equations and their representations through matrices and vector spaces. This field is foundational for many areas of mathematics and practical applications in engineering, physics, computer science, and economics.

Understanding the structure of vector spaces and the behavior of linear transformations lies at the heart of linear algebra. Concepts like basis vectors, which define the dimensionality and direction within a vector space, and matrix operations correlating to transformations of these vectors, are key to both theoretical and practical applications.

In Grassmann's combinatory multiplication, the interplay between the basis vectors expressed through linear combinations and the structure of a matrix determinant provides an example of these core linear algebra principles at work. It emphasizes the importance of grasping these concepts for solving complex mathematical problems.

Antisymmetric Properties

Antisymmetry is a property that arises in certain mathematical contexts, notably in the multiplication of basis vectors in a vector space. An antisymmetric operation, such as the cross product in a three-dimensional space, satisfies the condition that flipping the order of the operands results in the negation of the original outcome (for distinct operands) and equals zero when the operands are identical.

In the case of Grassmann's multiplication, the antisymmetric property ensures that when any two indices are the same, the product is zero, whereas if you swap two different indices, the sign of the product changes. This property underpins the simplification process in Step 3 and Step 4 of the Grassmann's combinatory multiplication proof, allowing the removal of terms where indices repeat and the guarantee that only distinct permutations contribute to the final sum.

Recognizing and applying antisymmetric properties is a valuable skill in linear algebra, as it is essential in simplifying expressions and contributes to a deeper understanding of the relationships between algebraic entities.