Hello all. Now i have stopped trying to visualize tensors and approach them from the roots i have made some progress but still have problems. I understand a tensor to be a multilinear mapping from vector spaces and their duals to the reals ( the underlying field ). How does a tensor act upon a vector to produce another vector. This seems to go against the definition. The same query applies when there are not the same number of vectors and one forms in the mapping as they cannot all be contracted to get a scalar or real number. Metheinste.