# Convert from one tensor canonical form to another

Suppose we have two canonical forms $$A, B \in \mathbb{F}_2^{2 \times 2 \times 2}$$ of a 3-dimensional tensor product space over the finite field with two elements, where

$$A = e_1 \otimes e_2 \otimes e_1 + e_2 \otimes e_1 \otimes e_1$$ and $$B = e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_2$$,

and $$e_1, e_2$$ are the usual standard basis of a 2-d vector space. Is it possible to "convert" from A to B? I am wondering if there are transformations that convert A to B (and vice versa)? Both $$A$$ and $$B$$ have the same tensor rank (i.e. they both have rank-2 because they can be written as the sum of two rank-1 tensors), so is this transformation rank-preserving? And if, say, I had two canonical forms that were not rank-equivalent, is there a transformation that takes me from one canonical form to another? Thank you.

If you remove the origin, every vector space is homogenous, meaning for any two nonzero vectors $$u, v$$ there is an invertible linear map $$A$$ with $$Au=v$$. So the answer is yes.