I Simplicity of tensor products?!

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I was just watching a video that was reviewing some linear algebra, and it said that this was the tensor product:

Let's say you have a matrix A and a matrix B (both 2 by 2 matrices). If I want to calculate the tensor A ⊗ B, then the answer is basically just a matrix of matrices. In other words, I do this:

The first matrix of the tensor product space is: the scalar multiplication of A11 * B
The 2nd matrix of the tensor product space is: A12 * B
The 3rd matrix is: A21 * B
The last matrix is: A22 * B

Over all, this makes a 4 by 4 matrix (which I will call C even though I know it should really be denoted A ⊗ B) with elements:

C11 = A11 * B11
C12 = A11 * B12
C13 = A12 * B11
C14 = A12 * B12
C21 = A11 * B21
C22 = A11 * B22
C23 = A12 * B21
C24 = A12 * B22

C31 = A21 * B11
C32 = A21 * B12
C33 = A22 * B11
C34 = A22 * B12
C41 = A21 * B21
C42 = A21 * B22
C43 = A22 * B21
C44 = A22 * B22

I just want to ask: Is this really all there is to taking a tensor product? Is this really the process or is this just some simplified special case? I just ask this because I have asked on threads before about tensor products and tried to look up videos and web pages on them, and every time my source has just made it out to be some daunting process that was so difficult to explain and just about impossible to show an example of.
 

fresh_42

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In coordinate form, your example is correct. An example of the tensor product of two matrices. In general there may be also sums of them to get other elements in the vector space of here ##\mathbb{M}_{2 \times 2} \otimes \mathbb{M}_{2 \times 2}##.

Edit: One can arrange them in different ways, e.g. as four layers of ##2 \times 2## matrices: ##A_{11}B ,...##
 

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