👉 The math behind linear transformations involves understanding how matrices represent these transformations in a coordinate system. A linear transformation \( T: \mathbb{R}^n \to \mathbb{R}^m \) can be described by a matrix \( A \), where each column of \( A \) corresponds to the image of the standard basis vectors of \( \mathbb{R}^n \) under \( T \). Specifically, if \( e_1, e_2, \ldots, e_n \) are the standard basis vectors in \( \mathbb{R}^n \), then the transformation \( T(e_1) = a_{11} e_1 + a_{12} e_2 + \cdots + a_{1n} e_n \), \( T(e_2) = a_{21} e_1 + a_{22} e_2 + \cdots + a_{2n} e_n \), and so on, can be expressed as \( T(e_i) = A_i \). This matrix \( A \) encapsulates all the information about how \( T \) acts on vectors in \( \mathbb{R}^n \), allowing us to compute the image of any vector in \( \mathbb{R}^n \) under \( T \) by multiplying it with \( A \). The properties of matrix multiplication, such as distributivity and associativity, ensure that the transformation behaves consistently across different coordinate systems.
