The core principle establishes a one-to-one correspondence between matrix transformations and linear transformations in finite-dimensional vector spaces. This theorem proves that any transformation defined by the algebraic operation $T(\mathbf{x}) = A\mathbf{x}$ satisfies the axioms of linearity (preservation of scalar multiplication and vector addition), and conversely, every function satisfying these linearity axioms can be represented as matrix-vector multiplication relative to a standard basis. This equivalence bridges the geometric interpretation of column space operations with the algebraic properties defining linear maps within the domain of Linear Algebra.
The core principle establishes a one-to-one correspondence between matrix transformations and linear transformations in finite-dimensional vector spaces. This theorem proves that any transformation d…