A4 - Injectivity and surjectivity

Example 1 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 5 & 0 \\ 0 & 1 & 0 \\ 0 & -4 & 1 \\ 1 & 7 & -3 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 5 & 0 \\ 0 & 1 & 0 \\ 0 & -4 & 1 \\ 1 & 7 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 2 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} -1 & 0 & 2 & 1 \\ 2 & 3 & -7 & 1 \\ -2 & -1 & 5 & 1 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 0 & 2 & 1 \\ 2 & 3 & -7 & 1 \\ -2 & -1 & 5 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -1 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 3 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 4 & 3 \\ 0 & 1 & 0 \\ 1 & 3 & 4 \\ -1 & 1 & -8 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 4 & 3 \\ 0 & 1 & 0 \\ 1 & 3 & 4 \\ -1 & 1 & -8 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 4 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} -1 & 3 & 3 \\ 0 & 1 & 1 \\ -2 & 7 & 7 \\ -2 & 5 & 5 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} -1 & 3 & 3 \\ 0 & 1 & 1 \\ -2 & 7 & 7 \\ -2 & 5 & 5 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 5 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & -4 & -3 \\ 0 & 1 & 1 \\ 1 & -8 & -7 \\ -1 & 8 & 7 \\ -2 & 5 & 3 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & -4 & -3 \\ 0 & 1 & 1 \\ 1 & -8 & -7 \\ -1 & 8 & 7 \\ -2 & 5 & 3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 6 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 0 & -1 & -1 & -3 \\ 1 & -1 & 0 & -3 \\ 0 & -1 & -1 & -3 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & -1 & -1 & -3 \\ 1 & -1 & 0 & -3 \\ 0 & -1 & -1 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 7 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} -2 & -2 & -3 & 3 \\ -5 & 1 & -4 & -4 \\ -3 & -2 & -4 & 3 \\ 5 & -2 & 4 & 4 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & -2 & -3 & 3 \\ -5 & 1 & -4 & -4 \\ -3 & -2 & -4 & 3 \\ 5 & -2 & 4 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. \(T\) is injective.
  2. \(T\) is surjective.

Example 8 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & -1 & -7 & 0 \\ 0 & 1 & 3 & -4 \\ -1 & 1 & 8 & 0 \\ 0 & -1 & 2 & 5 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -7 & 0 \\ 0 & 1 & 3 & -4 \\ -1 & 1 & 8 & 0 \\ 0 & -1 & 2 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. \(T\) is injective.
  2. \(T\) is surjective.

Example 9 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} -1 & 2 & 3 \\ -1 & 6 & 4 \\ 2 & -7 & -7 \\ 1 & -7 & -5 \\ 0 & -4 & 2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} -1 & 2 & 3 \\ -1 & 6 & 4 \\ 2 & -7 & -7 \\ 1 & -7 & -5 \\ 0 & -4 & 2 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 10 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & -2 & -7 & -8 \\ 0 & 1 & 2 & 3 \\ 0 & -1 & -1 & -2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -7 & -8 \\ 0 & 1 & 2 & 3 \\ 0 & -1 & -1 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{array}\right] \]

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

Example 11 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} -1 & -7 & 1 \\ 1 & 4 & -1 \\ -1 & -5 & 1 \\ 0 & 5 & 0 \\ -1 & -7 & 1 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} -1 & -7 & 1 \\ 1 & 4 & -1 \\ -1 & -5 & 1 \\ 0 & 5 & 0 \\ -1 & -7 & 1 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 12 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 0 & 2 \\ -2 & -7 & 6 \\ 0 & 2 & -3 \\ -1 & -3 & 2 \\ 2 & 7 & -2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & 2 \\ -2 & -7 & 6 \\ 0 & 2 & -3 \\ -1 & -3 & 2 \\ 2 & 7 & -2 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 13 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & 0 & -4 & 1 \\ -1 & 1 & 7 & 2 \\ -2 & -1 & 6 & -5 \\ 0 & 0 & 3 & 0 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 0 & -4 & 1 \\ -1 & 1 & 7 & 2 \\ -2 & -1 & 6 & -5 \\ 0 & 0 & 3 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 14 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & -1 & 2 & -3 \\ -1 & 2 & -7 & 1 \\ -2 & 2 & -3 & 7 \\ -1 & 1 & -7 & -2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 2 & -3 \\ -1 & 2 & -7 & 1 \\ -2 & 2 & -3 & 7 \\ -1 & 1 & -7 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 15 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} -3 & 5 & -1 \\ 0 & 1 & -3 \\ 1 & -1 & 2 \\ 2 & -2 & -2 \\ -2 & -2 & 7 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} -3 & 5 & -1 \\ 0 & 1 & -3 \\ 1 & -1 & 2 \\ 2 & -2 & -2 \\ -2 & -2 & 7 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 16 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 0 & -3 \\ 0 & 1 & -5 \\ -1 & 2 & -6 \\ -1 & 1 & 1 \\ 1 & -1 & -1 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & -3 \\ 0 & 1 & -5 \\ -1 & 2 & -6 \\ -1 & 1 & 1 \\ 1 & -1 & -1 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 17 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} -2 & 1 & 0 \\ -2 & -1 & -7 \\ 1 & 0 & 2 \\ -1 & 0 & -2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} -2 & 1 & 0 \\ -2 & -1 & -7 \\ 1 & 0 & 2 \\ -1 & 0 & -2 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 18 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 4 & 3 & 1 & -4 \\ 1 & 1 & 1 & 1 \\ 0 & -1 & -2 & -4 \\ -1 & 0 & 1 & 4 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 4 & 3 & 1 & -4 \\ 1 & 1 & 1 & 1 \\ 0 & -1 & -2 & -4 \\ -1 & 0 & 1 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. \(T\) is injective.
  2. \(T\) is surjective.

Example 19 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & -1 & -4 \\ 0 & 1 & 5 \\ 1 & -2 & -8 \\ -2 & 1 & 0 \\ 1 & 0 & 2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

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

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

Example 20 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} -1 & 0 & 1 \\ -2 & -1 & 4 \\ -2 & -2 & 6 \\ 0 & 3 & -6 \\ -5 & -1 & 7 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} -1 & 0 & 1 \\ -2 & -1 & 4 \\ -2 & -2 & 6 \\ 0 & 3 & -6 \\ -5 & -1 & 7 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 21 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 2 & 2 & -2 & 8 \\ -1 & -2 & 1 & -6 \\ 1 & -1 & -1 & 0 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & 2 & -2 & 8 \\ -1 & -2 & 1 & -6 \\ 1 & -1 & -1 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 22 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 4 & -6 \\ 1 & 6 & -8 \\ 0 & -1 & 1 \\ -1 & -6 & 8 \\ -1 & 0 & 2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 4 & -6 \\ 1 & 6 & -8 \\ 0 & -1 & 1 \\ -1 & -6 & 8 \\ -1 & 0 & 2 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 23 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 7 & -4 & 3 & 7 \\ 0 & 1 & 0 & -5 \\ 1 & -2 & 1 & 8 \\ -2 & 1 & -1 & -1 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 7 & -4 & 3 & 7 \\ 0 & 1 & 0 & -5 \\ 1 & -2 & 1 & 8 \\ -2 & 1 & -1 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. \(T\) is injective.
  2. \(T\) is surjective.

Example 24 🔗

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

Answer:

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

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

Example 25 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & -3 & 4 & -1 \\ 2 & -5 & 7 & -4 \\ -1 & -2 & 2 & 8 \\ 3 & -4 & 5 & -6 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -3 & 4 & -1 \\ 2 & -5 & 7 & -4 \\ -1 & -2 & 2 & 8 \\ 3 & -4 & 5 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. \(T\) is injective.
  2. \(T\) is surjective.

Example 26 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} -3 & 1 & -1 & 5 \\ -3 & 1 & -2 & 4 \\ -4 & 1 & 0 & 8 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -3 & 1 & -1 & 5 \\ -3 & 1 & -2 & 4 \\ -4 & 1 & 0 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{array}\right] \]

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

Example 27 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} -1 & -3 & 1 & -3 \\ 1 & 5 & 0 & 5 \\ -2 & -7 & 1 & -7 \\ 0 & -3 & 3 & -3 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -3 & 1 & -3 \\ 1 & 5 & 0 & 5 \\ -2 & -7 & 1 & -7 \\ 0 & -3 & 3 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 28 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} -1 & -1 & -1 & -2 \\ -2 & -3 & -2 & -6 \\ -3 & -1 & -2 & -4 \\ -1 & -4 & -3 & -4 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -1 & -1 & -2 \\ -2 & -3 & -2 & -6 \\ -3 & -1 & -2 & -4 \\ -1 & -4 & -3 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 29 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 0 & 5 & -5 \\ 1 & 4 & -2 \\ -1 & -6 & 4 \\ 1 & -2 & 4 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 0 & 5 & -5 \\ 1 & 4 & -2 \\ -1 & -6 & 4 \\ 1 & -2 & 4 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 30 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} -2 & 1 & 3 & 2 \\ 3 & -2 & -2 & -4 \\ -2 & 1 & 4 & 2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & 1 & 3 & 2 \\ 3 & -2 & -2 & -4 \\ -2 & 1 & 4 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 \end{array}\right] \]

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

Example 31 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & -1 & 6 & -6 \\ 0 & 1 & -3 & 3 \\ -1 & 2 & -8 & 8 \\ 2 & 0 & 6 & -6 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 6 & -6 \\ 0 & 1 & -3 & 3 \\ -1 & 2 & -8 & 8 \\ 2 & 0 & 6 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 32 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & -5 & 4 & -7 \\ 0 & 0 & 1 & -1 \\ 1 & -5 & 4 & -7 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -5 & 4 & -7 \\ 0 & 0 & 1 & -1 \\ 1 & -5 & 4 & -7 \end{array}\right] = \left[\begin{array}{cccc} 1 & -5 & 0 & -3 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 33 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 2 & 2 \\ 0 & 1 & 2 \\ 1 & 1 & 0 \\ 0 & -1 & -2 \\ 0 & 1 & 2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 2 & 2 \\ 0 & 1 & 2 \\ 1 & 1 & 0 \\ 0 & -1 & -2 \\ 0 & 1 & 2 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 34 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 0 & -4 & -4 \\ 0 & 5 & 5 \\ -1 & -5 & -6 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 0 & -4 & -4 \\ 0 & 5 & 5 \\ -1 & -5 & -6 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 35 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 1 & 0 \\ -3 & -2 & -2 \\ -1 & -3 & 4 \\ -3 & -4 & 2 \\ 0 & 2 & -4 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 1 & 0 \\ -3 & -2 & -2 \\ -1 & -3 & 4 \\ -3 & -4 & 2 \\ 0 & 2 & -4 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 36 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 3 \\ 1 & 1 & 2 \\ 0 & -1 & -3 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 3 \\ 1 & 1 & 2 \\ 0 & -1 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 37 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 5 & 0 \\ 1 & 6 & 0 \\ -1 & -6 & 0 \\ 0 & 5 & 0 \\ 0 & -1 & 0 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 5 & 0 \\ 1 & 6 & 0 \\ -1 & -6 & 0 \\ 0 & 5 & 0 \\ 0 & -1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 38 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & -2 & -7 & -1 \\ 0 & 1 & 2 & 0 \\ 1 & 3 & 3 & 0 \\ -4 & -4 & 4 & 0 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -7 & -1 \\ 0 & 1 & 2 & 0 \\ 1 & 3 & 3 & 0 \\ -4 & -4 & 4 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 39 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & -3 & -4 \\ 0 & 1 & 1 \\ 2 & -4 & -6 \\ 4 & -4 & -8 \\ 0 & 1 & 1 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & -3 & -4 \\ 0 & 1 & 1 \\ 2 & -4 & -6 \\ 4 & -4 & -8 \\ 0 & 1 & 1 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 40 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} 1 & -1 & -1 & 4 \\ 0 & 1 & 0 & -3 \\ -1 & 0 & 2 & -6 \\ -1 & 1 & 2 & -8 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -1 & 4 \\ 0 & 1 & 0 & -3 \\ -1 & 0 & 2 & -6 \\ -1 & 1 & 2 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  1. \(T\) is injective.
  2. \(T\) is surjective.

Example 41 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} -3 & 7 & 8 \\ -4 & 5 & 2 \\ 3 & 0 & 6 \\ 2 & -4 & -4 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} -3 & 7 & 8 \\ -4 & 5 & 2 \\ 3 & 0 & 6 \\ 2 & -4 & -4 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 42 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} -2 & -8 & 6 \\ -2 & -7 & 5 \\ -1 & -4 & 3 \\ 1 & 4 & -3 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} -2 & -8 & 6 \\ -2 & -7 & 5 \\ -1 & -4 & 3 \\ 1 & 4 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 43 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 0 & -4 \\ 0 & 1 & 5 \\ 0 & 1 & 6 \\ 2 & 2 & -3 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & -4 \\ 0 & 1 & 5 \\ 0 & 1 & 6 \\ 2 & 2 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 44 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} -1 & 1 & -3 & 2 \\ 2 & -1 & 4 & -4 \\ -2 & 1 & -3 & 3 \\ 2 & -2 & 3 & -1 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 1 & -3 & 2 \\ 2 & -1 & 4 & -4 \\ -2 & 1 & -3 & 3 \\ 2 & -2 & 3 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 45 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 4 & 7 \\ -2 & -1 & 6 \\ 0 & 1 & 5 \\ -2 & 0 & 7 \\ 0 & 3 & 8 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 4 & 7 \\ -2 & -1 & 6 \\ 0 & 1 & 5 \\ -2 & 0 & 7 \\ 0 & 3 & 8 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 46 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} -8 & 5 & 5 \\ 3 & -2 & -2 \\ -3 & 0 & 1 \\ -4 & 0 & 2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} -8 & 5 & 5 \\ 3 & -2 & -2 \\ -3 & 0 & 1 \\ -4 & 0 & 2 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 47 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} -5 & -3 & -4 & -3 \\ -3 & -2 & -2 & -3 \\ 1 & 2 & -1 & 7 \\ 0 & -2 & 1 & -6 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -5 & -3 & -4 & -3 \\ -3 & -2 & -2 & -3 \\ 1 & 2 & -1 & 7 \\ 0 & -2 & 1 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

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

Example 48 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & -2 & -2 \\ 2 & -3 & -2 \\ -3 & 7 & 8 \\ 1 & -5 & -8 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & -2 & -2 \\ 2 & -3 & -2 \\ -3 & 7 & 8 \\ 1 & -5 & -8 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 49 🔗

Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 5 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{ccc} 1 & 0 & 5 \\ 0 & 1 & -5 \\ 0 & -1 & 6 \\ 3 & 2 & 8 \\ 0 & 0 & 2 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & 5 \\ 0 & 1 & -5 \\ 0 & -1 & 6 \\ 3 & 2 & 8 \\ 0 & 0 & 2 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \]

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

Example 50 🔗

Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc} -2 & 3 & 1 & 8 \\ 1 & -2 & 1 & -5 \\ 0 & 2 & -5 & 4 \end{array}\right] .\)
  1. Explain why \(T\) is or is not injective.
  2. Explain why \(T\) is or is not surjective.

Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & 3 & 1 & 8 \\ 1 & -2 & 1 & -5 \\ 0 & 2 & -5 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 \end{array}\right] \]

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