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 $W^\alpha= 0$