Invariance of a tensor under coordinate transformation

Question

I know, that a tensor is a mathematically entity that is represented using a basis and tensor products, in the form of a matrix, and changing a representation doesn't change a tensor, is kind of obvious.

So does the invariance of a tensor under coordinate transformation mean what I stated above or does it mean that under a set of particular transformation the representation of a particular tensor also doesn't change.

Quoted from Wikipedia:

A vector is invariant under any change of basis, so if coordinates transform according to a transformation matrix $L$, the bases transform according to the matrix inverse $L^{−1}$, and conversely if the coordinates transform according to inverse $L^{−1}$, the bases transform according to the matrix $ L$.

Can someone please shed some light on this?

Answer 1

Regarding what you quoted: a vector is represented by the sum of a set of basis vectors times the vector components. If the components transform according to $L$, then the bases will transform according to $L^{-1}$, which means that when you multiply the bases with the components (to make the vector), you will get the same result every time (since $L\cdot L^{-1}=I$). This is what is meant by invariance.

Invariance of a tensor means basically what you stated above- the tensor itself doesn't change under a change of coordinates (like I explained). However, the tensor's components can very well change.

Answer 2

I guess there is two different notions of invariance of tensors. First notion is that if you look at a tensor as a mapping then the first notion of invariance is what you mentioned above. The other notion of invarance is that you do transformation but the " component" of metric does not change. For instance, if we do Lorentz transformation then the Minkowski metric is invariant, meaning that the component will be +1 , -1 , -1 ,-1 .

I might be wrong!!