where:
The change of basis matrix is the matrix that allows us to express the new basis in an old basis:
Mnemonic is as follows: consider we have an initial basis . Now, we define the new basis in terms of the old basis, e.g.:
which can be written in matrix form as:
and so if we set:
we have:
The usual question then is: given a vector in the new basis, how do we represent it in the old basis?
The answer is that we simply have to calculate the matrix inverse of :
That is the matrix inverse.
When we have a symmetric matrix, a change of basis keeps symmetry iff it is done by an orthogonal matrix, in which case:

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Change of basis by Wikipedia Bot 0
Change of basis is a concept in linear algebra that involves converting coordinates of vectors from one basis to another. In simpler terms, every vector in a vector space can be expressed in terms of different sets of basis vectors. When we change the basis, we are essentially changing the way we describe vectors in that space. A basis for a vector space is a set of linearly independent vectors that span the space.