Validity of Tensor Expressions in (a)-(d)

[Ibix]

Another trivial question from me.

Homework Statement

Which (if any) of the following are valid tensor expressions:
(a)$A^\alpha+B_\alpha$
(b)$R^\alpha{}_\beta A^\beta+B^\alpha=0$
(c)$R_{\alpha\beta}=T_\gamma$
(d)$A_{\alpha\beta}=B_{\beta\alpha}$

Homework Equations

Nothing relevant - these are generic tensors.

The Attempt at a Solution

(a) and (c) are not valid because the indices don't match up. (d) is valid - in matrix notation, A=B^T.

I'm not sure about (b), though. The left hand side is valid; summing over the dummy index makes it a sum of two vectors. I'm not quite sure how to interpret the equality, though. I can see it as a vector equaling a scalar - which is not valid. Alternatively, I can read an implicit $\forall \alpha$ - in other words, that each element of the vector on the left hand side is identically zero.

I lean towards the first interpretation - but I'm not sure.

 

[TSny]

For an equation like (b), the zero on the right would normally be interpreted as the zero vector (rather than the zero scalar); or more precisely, it would be the $\alpha$ component of the zero vector (which would of course have a value of 0 in any coordinate system and any reference frame). So, I think your second interpretation is better.

 

[Ibix]

Thank you very much.

This is self-study. When I downloaded the problem sheet the answers were available - now term has started and they've gone... Shouldn't have been conscientious.