A linear transformation is formally defined as a function mapping vectors to other vectors that preserves linearity by maintaining collinearity and fixing the origin at zero within Euclidean space. In two-dimensional domains, such transformations are characterized entirely by their effect on an orthonormal basis set, where every subsequent vector's image is determined via the superposition of linear combinations derived from these specific mappings. Consequently, matrices serve as finite representations encoding these transformation parameters in columns, and matrix-vector multiplication functions as the algebraic mechanism for computing images under such transformations without explicit geometric visualization.
A linear transformation is formally defined as a function mapping vectors to other vectors that preserves linearity by maintaining collinearity and fixing the origin at zero within Euclidean space. I…