Linear Maps are relations between vector spaces that preserve vector addition and scalar multiplication.
Appendix A
A linear relation between two second rank tensors means that each component of one tensor is a linear combination of all the components of the other tensor.
Elsewhere on these pages you were told that a component in a new vector depends on all the components of the old vector.
Change that to the following:
A component in this vector depends on all the components of that vector.
Vectoreverything 