Is it possible to consider/treat any linear application (INVERTIBLE, of course) in the same vector space, as an opportune change of basis?

I ask this because the starting point of the question is: how do we deal with the fact that a change of basis necessarily maintains the length of a vector unchanged, whilst a linear application... well it could not?

I mean, let's think about a vector: in this case we can think of it as a "horrible" arrow. If I make a transformation on this vector, it can expand, and have a different length. If I change the basis, this vector maintains the same lengths, without any doubt.

This is obvious both from a physical point of view, and because when I change the basis, the change does induce a change in the metrics such that the length will remain the same. BUT applying a linear transformation on a SINGLE vector, does not change the metric, hence its lengths could be different. Indeed, for a given metric $g$, I can find transformations that leave the lengths unchanged, transformations that satisfy

$$A^{T}gA = g$$

hence transformations such that the metric does remain the same under the change of coordinates given by their inverse).

BUT again if I consider a linear application that changes the whole space, then obviously it does remain all the same since being it an isomorphism, the two spaces are equal.

Any idea?

Thank you!


This statement

a change of basis necessarily maintains the length of a vector unchanged

from your question is false. The simplest counterexample is just one dimensional: both $\{1\}$ and $\{2\}$ are bases for $\mathbb{R}$. The change of basis matrix $[2]$ doubles lengths.

Any invertible linear transformation maps bases to bases - as you point out in your question.


I believe so. Because if you look at what the matrix of the linear transformation (rel the standard basis, say) does, it changes basis from the basis formed by the (necessarily linearly independent) columns of the associated matrix to the standard basis...

Although, in general such a transformation does not preserve the metric.


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