Do Vectors Remain Invariant and What Defines Tensor Rank?

[TupoyVolk]

"The components of a vector change under a coordinate transformation, but the vector itself does not."
ie:
V = a*x + b*y = c*x' + d*y'
Though the components (and the basis) have changed, V is still = V.
Question 1:
Is that right? (I'm assuming so, the main Q is below)

Tensor rank (according to wolfram)
"The total number of contravariant and covariant indices of a tensor."
It is commonly said
"A vector is a tensor of rank 1"

Does this mean (A):
T^a, and R_a
are tensors of rank one

or does it mean (B):
V = (T^a)(R_a) is a tensor of rank one?

If it is (A), then how can a vector be regarded as a tensor of rank 1, when it is
(contravariant components)*(covariant basis)

I'm able to do the maths, but the terminology of 'rank' has been bugging me!

 

[TupoyVolk]

Gah, I'm pretty sure from clicking the linked definition of tensor on here, I got the answer :P

Tensor of rank 1 = V^a*e_a

Components of a tensor of rank 1 = V^a.

Oui?

 

[RedX]

The components of a vector change when the basis is changed, but the vector does not, since a vector is something that exists without coordinates.

So your question 1 is right.

As for your other question, you don't count the basis when counting rank, just the indices on the component. If there are no indices on the components, then count the indices of the basis. But don't count both.

 

[TupoyVolk]

Cheers! Much appreciated.