Transformation
A transformation (or function or mapping) \(T\) from \(\mathbb{R^n} \mbox{ to } \mathbb{R^m}\)
Is a rule that assignms to each vector x in \(\mathbb{R^n}\) a vector \(T(\mathbf{x})\) in \(\mathbb{R^m}\)
Domain to CoDomain → \(T : \mathbb{R^n} \to \mathbb{R^m}\)
Matrix Transformation
\(\mathbf{x} \mapsto A\mathbf{x}\\ T : \mathbb{R^n} \to \mathbb{R^m}\)
Linear Transformation
A transformation (or mapping ) \(T\) is linear if
\(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\\
T(c\mathbf{u} = cT(\mathbf{u}))\) ( : Theorem 5.)
Every matrix transformation is a linear transformation : 모든 MT는 LT다. 매트릭스가 LT 정의를 그대로 따르기 때문에!