Proof that the transpose of a tensor is a tensor

In summary, the task is to prove that the transpose of a tensor is also a tensor. The definition of transpose states that a vector dotted with the transpose of another vector is equal to the transpose of that vector dotted with the first vector. However, in order to use this property, it must first be proven that the transpose of a tensor is a tensor. The definition of a tensor, which states that it is linear, does not seem to provide any useful information. Therefore, it may seem that one has to assume that T^T is a tensor and then prove it, but this is not the case. Further guidance or assistance would be appreciated.
  • #1
wonfish
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Homework Statement


Prove that the transpose of a tensor is a tensor.



Homework Equations


Definition of the transpose: a[itex]\bullet[/itex]Tb = b[itex]\bullet[/itex]T^Ta where a and be are arbitrary vectors



The Attempt at a Solution


This isn't homework per se, I'm 60 yo and studing continuum mechanics, but I see that I can't use any properties of transposes since I haven't proven first that T^T is a tensor. Likewise I can't use properties of tensors either except for the linear property defintion of a tensor : T(αa + βb) = αTa + βTb

This definition doesn't seem to get me anywhere.
 
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  • #2
It is almost as if I have to assume that T^T is a tensor and then prove it. I'm sure that's not the case but I don't know where to go from here.I would appreciate any help.
 

1. What is a tensor?

A tensor is a mathematical object that represents a multi-dimensional array of numbers or functions, which obey certain transformation rules under a change of coordinates. It is commonly used in physics and engineering to describe physical quantities such as displacement, velocity, and stress.

2. What does it mean for a tensor to be transposed?

Transposing a tensor means flipping its rows and columns, such that the elements in the first row become the elements in the first column, the elements in the second row become the elements in the second column, and so on. This operation can only be performed on tensors with two or more dimensions.

3. Why is it important to prove that the transpose of a tensor is a tensor?

Proving that the transpose of a tensor is a tensor is important because it confirms that the mathematical properties of tensors are preserved under transposition. This allows us to use tensors and their properties in various mathematical operations and calculations with confidence.

4. How is the transpose of a tensor calculated?

To calculate the transpose of a tensor, we simply swap the indices of the tensor. For example, if we have a tensor T with elements Tij, the transpose of T would be denoted as TT and its elements would be Tji.

5. Can any tensor be transposed?

No, not all tensors can be transposed. Only tensors with two or more dimensions can be transposed, as the operation involves flipping rows and columns. Additionally, tensors must also satisfy certain transformation rules in order to be considered a tensor, and not all mathematical objects fulfill these rules.

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