Proving W^a=0 in Tensor Multiplication: A critical analysis

[PhyPsy]

This book says that if W^aX_a=0 and X_a is arbitrary, then I should be able to prove that W^a=0. I don't see how this is possible. This is the equivalent of the vector dot product, so if, say, X_a=(1,0,0,0), then W^a could be (0,1,1,1), and the dot product would be 1*0+0*1+0*1+0*1=0. Why would W^a have to be 0?

 

[lurflurf]

It means for any X_a not for some X_a. So in your vector example the only vector orthogonal to all vectors is the zero vector.

 

[HallsofIvy]

If X_\alpha can be any vector, it can be W_\alpha. If W^\alpha X_\alpha= 0 for X+_\alpha any vector then W^\alpha W_\alpha= 0 which immediately gives W^\alpha= 0