That's right. The dimensionality of the tensor space depends on both the rank of the tensor and the dimensionality of the underlying vector space. A rank-2 te

That's right. The dimensionality of the tensor space depends on both the rank of the tensor and the dimensionality of the underlying vector space. A rank-2 tensor based on 3-D physical space has 9 dimensions because there are 9 distinct pairs of the 3 underlying basis vectors. In general, a rank-N tensor on an M-dimensional space will have M^N dimensions.