Linear Algebra – Systems of linear equations – Linear transformations

Definition: A transformation \(T\) from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is a rule that assigns to each vector \(\mathbf{x}\) in \(\mathbb{R}^n\) a vector \(T(\mathbf{x})\) in \(\mathbb{R}^m\). The set \(\mathbb{R}^n\) is called the domain of \(T\) and \(\mathbb{R}^m\) is called the codomain of \(T\). Notation: \(T:\mathbb{R}^n\to\mathbb{R}^m\).
For \(\mathbf{x}\in\mathbb{R}^n\) the vector \(T(\mathbf{x})\) is called the image of \(\mathbf{x}\). The set of all images \(T(\mathbf{x})\) is called the range of \(T\).

Definition: A transformation \(T\) is called linear if

1. \(T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})\) for all \(\mathbf{u}\) and \(\mathbf{v}\) in the domain of \(T\);


2. \(T(c\mathbf{u})=cT(\mathbf{u})\) for all \(\mathbf{u}\) in the domain of \(T\) and all scalars \(c\in\mathbb{R}\).

Corollary: If \(T\) is a linear transformation, then \(T(\mathbf{0})=\mathbf{0}\) and \(T(c\mathbf{u}+d\mathbf{v}) =cT(\mathbf{u})+dT(\mathbf{v})\) for all \(\mathbf{u}\) and \(\mathbf{v}\) in the domain of \(T\) and all scalars \(c\) and \(d\) in \(\mathbb{R}\).

Generalization: \(T(c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots+c_p\mathbf{v}_p)=c_1T(\mathbf{v}_1)+c_2T(\mathbf{v}_2)+\cdots+c_pT(\mathbf{v}_p)\).

Matrix transformations: If \(A\) is an \(m\times n\) matrix, then the transformation \(T:\mathbb{R}^n\to\mathbb{R}^m,\; T(\mathbf{x})=A\mathbf{x}\) is a linear transformation.

Proof: For all \(\mathbf{u}\) and \(\mathbf{v}\) and for all scalars \(c\in\mathbb{R}\) we have:

\[T(\mathbf{u}+\mathbf{v})=A(\mathbf{u}+\mathbf{v})=A\mathbf{u}+A\mathbf{v}=T(\mathbf{u})+T(\mathbf{v})\quad\text{and}\quad T(c\mathbf{u})=A(c\mathbf{u})=cA\mathbf{u}=cT(\mathbf{u}).\]

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Last modified on March 22, 2021