Let
\[\mathbf{A} = \left[\begin{array} {rrrr} 1 & 2 & 1 & 1 \\ 2 & 1 & 1 & 0 \\ 1 & 2 & 1 & 2 \\ 1 & 2 & 1 & 1 \end{array}\right]\]
and let \(T:C^{4} \to C^{4}\) be given by \(T(x) = Ax\). Find \(\kappa(T)\). Is T injective?
Let us RREF the matrix:
\[\mathbf{A} = \left[\begin{array} {rrrr} 1 & 2 & 1 & 1 \\ 0 & -3 & -1 & -2 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]
\[\mathbf{A} = \left[\begin{array} {rrrr} 1 & 2 & 1 & 1 \\ 0 & 1 & 1/3 & 2/3 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]
\[\mathbf{A} = \left[\begin{array} {rrrr} 1 & 0 & 1/3 & 0 \\ 0 & 1 & 1/3 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]
Solving this set of equations yields:
\[x_{1} = -\frac{1}{3}x_{3}\]
\[x_{2} = -\frac{1}{3}x_{3}\] \[x_{4} = 0\]
Therefore, \[\kappa(T) = \langle \begin{bmatrix}-1\\ -1\\ 3\\ 0 \end{bmatrix} \rangle\]
The kernel in our case is non-trivial, so this transformation cannot be injective.