How do I make a change of basis with tensors in multilinear algebra?

In summary, during linear algebra studies, the process of changing between foreign bases and the standard basis was learned. This involves multiplying the change of basis matrix by the vector in coordinates with respect to the foreign basis, resulting in the vector in coordinates with respect to the standard basis. When changing between two different foreign bases, the formula remains the same, but the foreign basis is substituted for the standard basis. Changing bases with tensors involves deriving basis vectors for a coordinate system, creating a change of basis matrix, and multiplying it by the tensor in the original basis to get the tensor in the desired basis. A good resource for learning about multilinear algebra and tensor analysis is the linear algebra video playlist on mathispower4u.
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I did some linear algebra studies and learned how to change between foreign bases and the standard basis:

Change of basis matrix multiplied by the vector in coordinates with respect to the foreign basis equals the vector in coordinates with respect to the standard basis.

Of course, this is just changing bases while operating with vectors. Now I have a few questions:

1. What if I want to change between two different foreign bases (by foreign I just mean as opposed to the standard basis)? Is the formula still the same except you just put the foreign basis you are trying to get to in place of the standard basis in the formula I listed above?

2. How do you make a change of bases with tensors? My guess would be that you derive the basis vectors for a coordinate system, put those basis vectors in a change of basis matrix and multiply this change of basis matrix by the tensor in the basis you are trying to get away from. This should equal that same tensor in the basis you are trying to get to. Please correct me if I am wrong.

3. Where can I find a good multi linear algebra video playlist? Khan academy has linear algebra, but its playlist doesn't include multilinear algebra or tensor analysis of any sort. I found a play list on youtube, but all the videos were made private, so I can't see them.
 
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  • #2
Try this list from mathispower4u

https://dl.dropboxusercontent.com/u/28928849/Webpages/LinearAlgebraVideoLibraryTable.htm

It doesn't cover tensor notation though but might answer your basis questions.
 

FAQ: How do I make a change of basis with tensors in multilinear algebra?

What is a change of basis in multilinear algebra?

A change of basis in multilinear algebra refers to the process of transforming a tensor from one basis to another. This is necessary when working with tensors because different bases can represent the same tensor in different ways, and it is often useful to be able to switch between bases to simplify calculations or gain new insights.

How do I determine the transformation matrix for a change of basis?

The transformation matrix for a change of basis can be determined by performing a change of basis on the basis vectors themselves. This involves expressing the new basis vectors in terms of the old basis vectors, and then constructing a matrix with these new basis vectors as columns. This transformation matrix can then be used to convert between the two bases.

What is the relationship between the transformation matrix and the tensor components in a new basis?

The transformation matrix is used to convert the tensor components from one basis to another. Specifically, to convert the components from the old basis to the new basis, the transformation matrix is multiplied by the old components. This results in the new components in the new basis.

Can a change of basis be applied to any type of tensor?

Yes, a change of basis can be applied to any type of tensor, including scalar, vector, and higher-order tensors. However, the transformation process may differ slightly depending on the type of tensor, as each type has its own unique properties and transformation rules.

Are there any special considerations to keep in mind when making a change of basis with tensors?

Yes, there are a few important considerations to keep in mind when performing a change of basis with tensors. These include ensuring that the transformation matrix is invertible, understanding how contravariant and covariant components transform differently, and being aware of any symmetry or special properties of the tensor and how they may be affected by the change of basis.

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