TITLE: Is Geometric Algebra isomorphic to Tensor Algebra?
QUESTION [17 upvotes]: Is geometric algebra (Clifford algebra) isomorphic to tensor algebra?  If so, how then would one relate a unique 2-vector (this is what I'm going to call a multivector that is the sum of a scalar, vector, and bivector) to every 2nd rank tensor?

Edit by the OP, converted from "answer"
Okay.  Well I'm still curious if there's a way to represent any 2nd-rank tensor by a bivector, vector, and scalar.  Or in particular, can any $3 \times 3$ matrix be represented by a 2-vector in 3D.
It seems to me that they can't because I would guess the matrix representation of a bivector (grade 2 element of a 2-vector) would be exactly the same as the $3 \times 3$ matrix representation of the cross product (i.e. $[a \times b]_{ij} = a_j b_i - a_i b_j$) which only uniquely identifies 3 components.
I would also assume that the scalar part of the 2-vector would be represented by a scalar times the $3 \times 3$ identity matrix.  This would fill in 3 numbers, but really only uniquely gives 1 component.
I don't know how to represent the vector component of the 2-vector as a $3 \times 3$ matrix but I don't see how it could identity the remaining 5 components by itself.
Am I right then in assuming that there is a canonical matrix representation of a general 2-vector, but that there are matrices that cannot be represented by any 2-vector?

REPLY [4 votes]: Similar question: what's the relationship of tensor and multivector
The short answer is that all multivectors are tensors, but not all tensors are multivectors, so geometric algebra is not and cannot be isomorphic to tensor algebra.
However, we can try to embed geometric algebra inside of tensor algebra. These resources give some guidelines for doing so (in the special case where the vector space is $\mathbb{R}^3$ and the inner product is the "regular" inner product, as opposed to say the Minkowski metric on $\mathbb{R}^4$):
http://www2.ic.uff.br/~laffernandes/teaching/2013.1/topicos_ag/lecture_18%20-%20Tensor%20Representation.pdf
https://www.docdroid.net/uwfvUxE/tensor-representation-of-geometric-algebra.pdf.html