Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & -1 & 0 & 5 \\ 0 & 1 & -2 & -1 \\ 0 & -4 & 8 & 5 \\ 0 & -1 & 2 & -2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 0 & 5 \\ 0 & 1 & -2 & -1 \\ 0 & -4 & 8 & 5 \\ 0 & -1 & 2 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  1. \(T\) is not injective
  2. \(T\) is not surjective