Would a set of Cartesian Tensors of rank 2 in 3 dimensions form a linear vector space?

Question

I understand that square matrices do form linear vector spaces. Although a tensor of rank 2 and a square matrix are not exactly the same thing, tensors too should have same addition properties at least in Cartesian coordinates. So can we say that tensors of rank 2 in 3D Cartesian coordinates consitute a linear vector space?

Answer 1

Yes, they do form a linear vector space. In fact, tensors of any rank on any dimensional vector space form a linear vector space. i.e given a linear vector space $V$ (say over $\Bbb{R}$), the space $T^r_s(V)$ of $(r,s)$ tensors over $V$ is indeed a linear vector space (over the field $\Bbb{R}$).

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By the way, a tensor is a thing which can be defined without respect to any particular coordinate system.