Find the matrix representation of: T: \(T: C^3 -> C^4, T \left(\begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} 3x + 2y +z \\ x + y + z \\ x-3y \\ 2x + 3y + z \end{bmatrix}\)
The matrix on the right side can be written as: \(\begin{bmatrix} 3x \\ x \\ x \\ 2x \end{bmatrix} + \begin{bmatrix} 2y \\ y \\ -3y \\ 3y \end{bmatrix} + \begin{bmatrix} z \\ z \\ 0 \\ z \end{bmatrix}\)
\(= x\begin{bmatrix} 3 \\ 1 \\ 1 \\ 2 \end{bmatrix} + y\begin{bmatrix} 2 \\ 1 \\ -3 \\ 3 \end{bmatrix} + z\begin{bmatrix} 1 \\ 1 \\ 0 \\ 1 \end{bmatrix}\)
\(= \begin{bmatrix} 3 & 2 & 1 \\ 1 & 1 & 1 \\ 1 & -3 & 0 \\ 2 & 3 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}\)
We know that matrices build linear transformations. Theorem \(MBLT\) states that if function T maps \(C^n\) to \(C^m\) by \(T(x) = Ax\) then T is a linear transformation represented by the matrix A. We have proved that T can be represented as the multiplication of a matrix A and x. The transformation T can be represented as the matrix:
\(\begin{bmatrix} 3 & 2 & 1 \\ 1 & 1 & 1 \\ 1 & -3 & 0 \\ 2 & 3 & 1\end{bmatrix}\)