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.

 

[blue_raver22]

Yes, it is correct that the components of a vector change under a coordinate transformation, but the vector itself does not. This is because a vector is defined by its magnitude and direction, which do not change under a transformation.

In terms of tensor rank, a vector can be considered a tensor of rank 1 because it has one contravariant index and zero covariant indices. In other words, it can be represented by a single column of numbers. In your example, both T^a and R_a would be considered tensors of rank 1 because they each have one index. However, when we multiply them together, we are creating a new tensor of rank 2 because the resulting tensor will have two indices (one from each of the original tensors).

The terminology of rank in tensor analysis can be confusing, but it is important to remember that it refers to the number of indices a tensor has, not the number of components. A tensor of rank 1 can have multiple components, as in the case of a vector, but it still only has one index. I hope this helps clarify the concept for you.