Let \(V\) and \(W\) be vector spaces. The function \(T: V \rightarrow W\) is a linear transformation of \(V\) into \(W\) when the two properties below are true for all \(\mathbf{u}\) and \(\mathbf{v}\) in \(V\) and for any scalar \(c\text{,}\)
\begin{equation*}
T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})\quad\text{ and } \quad T(c \mathbf{u})=c T(\mathbf{u}).
\end{equation*}
Example7.1.1.
Show that the map \(T:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}\) defined by
Compute all polynomials \(f(x)=x^{2}+bx+c\) such that \(T(f(x))=3f(x)\)
To study the linear transformation, we only need to know the image of a basis vector of \(V\text{.}\)
Theorem7.1.5.
Let \(T\) be a linear transformation from \(V\) to \(W\text{,}\) and \(\{\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n\}\) be a basis of \(V\text{.}\) Then \(T\) is totally determined the image of \(\{\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n\}\text{.}\) That is, if knowing \(T(\mathbf{v}_1),T(\mathbf{v}_2),\ldots,T(\mathbf{v}_n)\text{,}\) then we know \(T(v)\) for any vector \(v\in V\text{.}\)
Proof.
\(\forall v\in V\text{,}\) since \(\{\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k\}\) is a basis of \(V\text{,}\)
\begin{equation*}
v = x_{1}\mathbf{v}_{1}+x_{2}\mathbf{v}_{2}+\ldots+x_{n}\mathbf{v}_{n}
\end{equation*}
Let \(T\) be a linear transformation from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{m}\text{,}\) and \(\{\mathbf{e}_1,\mathbf{e}_2,\ldots,\mathbf{e}_n\}\) be the standard basis of \(\mathbb{R}^{n}\text{.}\) Then \(\forall v\in V,\)
\begin{equation*}
T(v)=Av,
\end{equation*}
where \(A=\Big[T(\mathbf{e}_1)\, T(\mathbf{e}_2)\,\ldots\, T(\mathbf{e}_n)\Big]\text{.}\)
The matrix \(A\) is called the standard matrix of \(T\text{.}\)
Example: Let \(T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}\) be a linear transformation such that
