- Is the function below a linear transformation? Why or why not?
\begin{equation*}
\ltdefn{T}{\complex{3}}{\complex{2}},\quad
\lt{T}{\colvector{x_1\\x_2\\x_3}}=\colvector{3x_1-x_2+x_3\\8x_2-6}
\end{equation*}
- Determine the matrix representation of the linear transformation $S$ below.
\begin{equation*}
\ltdefn{S}{\complex{2}}{\complex{3}},\quad
\lt{S}{\colvector{x_1\\x_2}}=\colvector{3x_1+5x_2\\8x_1-3x_2\\-4x_1}
\end{equation*}
-
Theorem LTLC has a fairly simple proof. Yet the result itself is very powerful. Comment on why we might say this.