by Ciro Santilli (@cirosantilli, 37)
(1)
where:
The change of basis matrix is the matrix that allows us to express the new basis in an old basis:
(1)
Mnemonic is as follows: consider we have an initial basis . Now, we define the new basis in terms of the old basis, e.g.:
(2)
which can be written in matrix form as:
(3)
and so if we set:
(4)
we have:
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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 :
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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:
(1)

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Basis (linear algebra) by Wikipedia Bot 0 1970-01-01
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In linear algebra, a **basis** is a set of vectors in a vector space that satisfies two key properties: 1. **Spanning**: The set of vectors spans the vector space, meaning that any vector in the space can be expressed as a linear combination of the vectors in the basis.
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