Tensor Analysis and Linear Algebra, what's the difference

[mahinda]

Hi,

I think I have been having this question for some time now. What is the difference between tensor calculus and linear algebra? Both seem to make frequent use of matrices, but they seem to be different subject matter. Can anyone please enlighten me on this issue?

 

[closet mathemetician]

Hi,

I think I have been having this question for some time now. What is the difference between tensor calculus and linear algebra? Both seem to make frequent use of matrices, but they seem to be different subject matter. Can anyone please enlighten me on this issue?

Hi Mahinda,

I'm no expert on either subject, but I've had the same question. I'll attempt to answer it. I think tensor calculus and linear algebra are two different ways of looking at some similar subject matter. Tensor calculus is more like an extension of vector calculus. Calculating derivatives and integrals of multi-variable functions in a more traditional calculus approach.

Linear algbegra takes a less visual, less,geometric approach and treats everything as sets and groups of objects which can be acted on by linear functionals in the form of matrices.

There is definitely overlap in the subject matter.

 

[slider142]

There is indeed a lot of overlap, as tensors are multilinear maps.
In linear algebra you study properties of linear maps between vector spaces.
You can then apply your knowledge of linear algebra to your study of tensors, as they are an ordered list of linear maps between many potentially different vector spaces: a multilinear map.
They arise naturally in differential geometry as a generalization of locally linear maps (as derivatives become in vector calculus).