Since affine transforms involve a matrix, if the transform matrix is a tensor, it would be of rank two. But, the real question is whether or not a change of basis, or transformation of the underlying space, effects the resultant vector linearly. Is that the correct argument? If the previous is true, I'd argue the affine transformation matrices are tensors since affine transformations are also linear in nature.

So are affine transformation matrices tensors of rank two? Similar to the rank two stress tensor composed of matrices.

This question is motivated by a desire to find intuitive examples of tensors, since I already know what an IFS is, this would make things very easy.

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I am going to suppose that by affine transformation you mean a mapping from $T : V \to V$ from a vector space to itself s.t. $$T(x) = Ax + b$$ where $A : V \to V$ is a linear map, and $b$ is an element of the vector space.

If that is the case, then no $T$ is not a tensor. A tensor needs to be a linear map on the tangent space of each point of the space on which it is defined. To make that statement concrete, I will take e.g. $V = \mathbb{R}^n$. Then we could consider $T$ as the same mapping acting at each point of $\mathbb{R}^n$ on its tangent space which is also $\mathbb{R^n}$. As a mapping on the tangent space it is given by $$T(x) = Ax + b$$ which is not linear.

Note however that the mapping $A$ from the above is a tensor.

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  • $\begingroup$ Awesome, pretty much what I expected. $\endgroup$ – Zach466920 Jun 21 '15 at 22:45

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