Problem: Transformation of coordinates under change of basis
Linear Algebra
Let
\(\mathbf{e}_1 , \mathbf{e}_2, \cdots, \mathbf{e}_n \,, \, \, \text{and} \,\,\, \, \mathbf{e}_1^{\prime} , \mathbf{e}_2^{\prime}, \cdots, \mathbf{e}_n^{\prime}
\)
be two bases of an n-dimensional vector space. Let these two bases be related via a square transformation matrix:
\(\mathbf{e}^{\prime} = \mathcal{A} \, \mathbf{e}\)
with a non-null determinant.
We can express any vector of this vector space as a linear combination of either basis above. How are the coordinates of a generic vector in these two bases related?