Zero tensor is the same for every coordinate system (proof)

Question

The tranformation formula for tensors is $$ T^{'}_{i_1,i_2,\dots,i_n}=a_{i_1j_1}a_{i_2j_2}\dots a_{i_kj_k}T_{j_1j_2\dots j_n} $$ Let $T_{j_1j_2\dots j_n}$ be the zero tensor. If we call $A$ the matrix of direction cosines and $T^{'}$ is $T$ rotated, in an arbitrary coordinate system. How can I show that $T^{'}$ which given by $$T^{'}_{i_1,i_2,\dots,i_n}=AT_{j_1j_2\dots j_n} $$ is also the zero tensor?

Answer 1

The components of the tensor in the new basis are linear combinations of the components in the old basis. The components in the old basis are all zero. So the components in the new basis will be a linear combination of a bunch of zeros. Thus they will be zero.