Multi-linear algebra Dual basis

In summary, the conversation is about proving exercise 4.1(a) from Spivak, where the goal is to show that a certain tensor product is equal to 1. The approach used involves using the dual basis and the alternating property to simplify the calculation. The question asked is whether this approach is the easiest way to solve the problem and whether it can be applied to all tensors. The conclusion is that the second equals sign in the given equation is true and can be applied to all tensors, but it may be more obvious for some tensors than others.
  • #1
brydustin
205
0
{(a_i)_j} is the dual basis to the basis {(e_i)_j}
I want to show that
((a_i)_1) \wedge (a_i)_2 \wedge... \wedge (a_i)_n ((e_i)_1,(e_i)_2,...,(e_i)_n) = 1

this is exercise 4.1(a) from Spivak. So my approach was:

\BigWedge_ L=1^k (a_i)_L ((e_i)_1,...,(e_i)_n) = k! Alt(\BigCross_L=1^k (a_i)_L)((e_i)_1,...,(e_i)_n)= k! Alt(T)((e_i)_1,...,(e_i)_n) = k!(1/k! Sum _ {permutations σ} sgn σ T ((e_i)_σ (1),...,(e_i)_σ (n))

where T = \BigCross_L=1^k (a_i)_L

So there is already a result on what T ((e_i)_1,...,(e_i)_n) is. 1 if all the sub-indices agree, and 0 otherwise. My question is... is T ((e_i)_σ (1),...,(e_i)_σ (n)) any different?

I'm assuming that in the one dimensional case we would say that T acts on one element in a linear fashion... but I'm kinda confused by the idea of having several arguments...

Otherwise,...is there an easier approach to the solution?
 
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  • #2
Okay... I have since figured out the solution ... the real question then becomes why is the second equals sign true (below):

×_L=1^k (a_i)_L ((e_i)_σ(1),... (e_i)_σ(k)) = ∏_L=1^k ((a_i)_L)(e_i)_σ(L) = 1 if and only if is identity and 0 otherwise.

where × denotes multiple(indexed) tensor products. And if this second equals sign is true then can I have this view for all tensor products? Namely, is a tensor product of k-parts operating on k-arguments equal to the product of each "part" acting on its corresponding argument (with the same index)?
Can I always hold that view of a tensor? Are there tensors where this is more obvious and others ... not so much?
 

1. What is multi-linear algebra dual basis?

Multi-linear algebra dual basis is a mathematical concept in linear algebra that involves the use of two bases, one for a vector space and another for its dual space. A basis is a set of vectors that can be used to represent any vector in a vector space. The dual space of a vector space is the set of all linear functionals on that vector space. Dual bases are used to define operations such as inner products and tensor products in multi-linear algebra.

2. How is a multi-linear algebra dual basis different from a regular basis?

A multi-linear algebra dual basis is different from a regular basis in that it consists of two sets of vectors, one for the vector space and one for its dual space, whereas a regular basis is just a single set of vectors for the vector space. Another difference is that the vectors in a dual basis are not necessarily orthogonal or normalized, as they are in a regular basis.

3. What is the purpose of using a multi-linear algebra dual basis?

The purpose of using a multi-linear algebra dual basis is to define operations such as inner products and tensor products in multi-linear algebra. These operations are essential for solving problems in fields such as physics, engineering, and computer science. Dual bases also allow for the representation of tensors, which are multidimensional arrays of numbers, in a compact and efficient manner.

4. How are dual bases related to matrices?

Dual bases are related to matrices through the use of the transpose operation. When a matrix is transposed, its rows and columns are interchanged. This allows for the representation of dual bases as matrices, with one set of vectors as rows and the other set as columns. The transpose of a matrix is also used to define the adjoint of a linear operator, which is a key concept in multi-linear algebra.

5. Can dual bases be used in other areas of mathematics besides multi-linear algebra?

Yes, dual bases can be used in other areas of mathematics, such as functional analysis and differential geometry. In functional analysis, dual bases are used to define the dual space of a topological vector space. In differential geometry, dual bases are used to define the tangent space and cotangent space of a manifold. They are also used in the study of differential forms, which are mathematical objects that generalize the concepts of length, area, and volume in higher dimensions.

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