Tensor Products and Associative Algebras

In summary: The rank of a tensor is the rank of the smallest matrix that can represent it. In other words, if we have a rank 2 tensor, we can represent it with a 2x2 matrix. But if we have a rank 5 tensor, we can't represent it with a 2x2 matrix, we need a 5x5 matrix.
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I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...

I am focused on Section 10.3 The Tensor Algebra ... ...

I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as follows:View attachment 5550
View attachment 5551In the above text from Cooperstein, in Definition 10.5, we read the following:

" ... ... An element \(\displaystyle x \in \mathcal{T}(V)\) is said to be homogeneous of degree \(\displaystyle d\) if \(\displaystyle x \in \mathcal{T}_d (V)\) ... ..."My question is as follows:

How can x be such that \(\displaystyle x \in \mathcal{T}(V)\) and \(\displaystyle x \in \mathcal{T}_d (V)\) ... does not seem possible to me ... ...

... ... because ... ...

... if \(\displaystyle x \in \mathcal{T}(V)\) then \(\displaystyle x\) will have the form of an infinite sequence as in the following:\(\displaystyle x = (x_0, x_1, x_2, \ ... \ ... \ , x_{d-1}, x_d, x_{d+1}, \ ... \ ... \ ... \ ... )
\)where \(\displaystyle x_i \in \mathcal{T}_i (V)\)... ... clearly \(\displaystyle x_d\) is the \(\displaystyle d\)-th coordinate of \(\displaystyle x\) and so cannot be equal to \(\displaystyle x\) ... ..

Can someone please clarify this issue ... clearly I am not understanding this definition ...

Peter
 
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The elements of $\mathcal{T}(V)$ are *finite* sequences (they are functions with finite support in $I$).

The elements that are homogeneous are tensors that are of the same "rank". For example, an element of $\mathcal{T}_3(V)$ might look like:

$v_1 \otimes v_2 \otimes v_3 + w_1 \otimes w_2 \otimes w_3$ where each $v_i, w_i \in V$.

It's analogous to the degree of a polynomial, where "homogeneous" of degree $d$ would mean all but the coefficients of $x^d$ are 0.

Remember the injections $\epsilon_i : \mathcal{T}_i(V) \to \bigoplus\limits_i \mathcal{T}_i(V)$? An element of $\mathcal{T}(V)$ is homogeneous if it is in the image of a single such injection.

Rank 0 tensors = scalars.
Rank 1 tensors = vectors.
Rank 2 tensors = 2-tensors, etc.

We're going to create a "giant algebra" of tensors of all ranks.
 

1. What are tensor products and how are they used in mathematics and science?

Tensor products are a mathematical operation that combines two vector spaces to create a new, larger vector space. They are used in mathematics and science to study multilinear algebra, differential geometry, and quantum mechanics, among other fields. Tensor products are also used in physics to describe the behavior of physical quantities and their transformations.

2. How are tensor products related to associative algebras?

Tensor products are closely related to associative algebras, as they can be used to construct and manipulate these algebraic structures. In particular, tensor products are used to define the multiplication operation in associative algebras, where the product of two elements is given by their tensor product. Additionally, the tensor product of two associative algebras is itself an associative algebra.

3. What is the significance of the associativity property in associative algebras?

The associativity property in associative algebras states that the order in which operations are performed does not affect the result. This property is important because it allows for simpler and more efficient calculations, as well as providing a clear and consistent structure for studying these algebras.

4. Can tensor products and associative algebras be applied to real-world problems?

Yes, tensor products and associative algebras have many practical applications in fields such as physics, engineering, and computer science. For example, they are used in signal processing, image recognition, and quantum computing. By understanding and utilizing these mathematical tools, scientists and engineers are able to solve complex problems and develop innovative technologies.

5. What are some common misconceptions about tensor products and associative algebras?

One common misconception is that tensor products and associative algebras are only relevant in pure mathematics and have no practical applications. As mentioned earlier, these concepts are used extensively in various fields of science and technology. Another misconception is that tensor products are only defined for vector spaces, when in fact they can be extended to more general mathematical structures such as modules and algebras.

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