Vectors are whatever satisfy the vector space axioms. Matrices are linear maps between vectors. Higher order tensors are multilinear maps. These are the standard conceptions of (multi)linear algebra.
But coming from a computational standpoint, is there any notion of a tensor as just grid-structured data? I.e., it is common to think of a grayscale image as a matrix where each element is a greyscale pixel value. In this case, the matrix is data and to think of it as a (linear) map just makes things confusing. Moreover, it is also natural to represent grayscale video data as a 3rd order tensor where each "frontal slab/slice" is a (matrix) frame of the video. To think of this as a multilinear map seems to suggest more than just a static blob of structured data, but the latter is all I want to work with.
So is there any notion of tensors as just static structured data blobs rather than as multilinear maps?