Linear Transformation and Matrix Similarity Under Different Bases

What is a Linear Transformation

A linear transformation is a function $ T: V \rightarrow V $ between vector spaces $ V $ and $ W $ that preserves vector addition and scalar multiplication. Formally, for any vectors $ u, v \in V $ and any scalar $ \alpha $, the transformation satisfies $ T(u + v) = T(u) + T(v) $ and $ T(\alpha u) = \alpha T(u) $. Such transformations can be represented by matrices, where the matrix $ A $ of $ T $ with respect to a particular basis is defined such that $ T(v) = A v $ for any vector $ v $. The choice of basis significantly influences the form of the matrix, and this is where the concept of similarity becomes crucial.

The Role of Bases in Linear Transformations

The representation of a linear transformation is inherently tied to the basis chosen for the vector space. Suppose $ {e_1, e_2, \dots, e_n} $ is a basis for $ \mathbb{R}^n $, and $ {f_1, f_2, \dots, f_n} $ is another basis. The matrix $ A $ of the transformation $ T $ with respect to the first basis is given by $ A = [T(e_1), T(e_2), \dots, T(e_n)] $. If the basis changes, the matrix representation of $ T $ transforms according to the change of basis formula:
$$ A' = P^{-1} A P, $$
where $ P $ is the matrix whose columns are the vectors of the new basis. This transformation, known as a change of basis, alters the form of the matrix but preserves the underlying linear transformation.

Matrix Representation and Change of Basis

The matrix $ A $ of a linear transformation $ T $ with respect to a basis $ {e_1, \dots, e_n} $ is a square matrix of size $ n \times n $. If $ P $ is the change of basis matrix from the original basis to the new basis, the new matrix $ A' $ is obtained via the equation $ A' = P^{-1} A P $. This relationship highlights that the transformation $ T $ is invariant under changes of basis, as the matrix representation adapts to the new coordinate system.

For example, consider a linear transformation $ T $ on $ \mathbb{R}^2 $ with matrix $ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} $ with respect to the standard basis. If the basis changes to $ { \begin{bmatrix} 1 \ 0 \end{bmatrix}, \begin{bmatrix} 0 \ 1 \end{bmatrix} } $, the matrix remains $ A $. However, if the basis is changed to $ { \begin{bmatrix} 2 \ 1 \end{bmatrix}, \begin{bmatrix} 1 \ 2 \end