An example of a linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ that has a non-trivial kernel is the following:

Transformation Definition:

Let $T$ be the transformation represented by the matrix: $T(x, y, z) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x \\ y \\ 0 \end{bmatrix}.$ This is a linear transformation that projects a vector $(x, y, z)$ in $\mathbb{R}^3$ onto the $xy$-plane by making the $z$-component zero.

Kernel of $T$:

The kernel of $T$, denoted by $\ker(T)$, consists of all vectors in $\mathbb{R}^3$ that are mapped to the zero vector by the transformation. In this case, $T(x, y, z) = (0, 0, 0)$ when: $\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ z \end{bmatrix}.$ Thus, the kernel is the set of all vectors of the form $(0, 0, z)$, which is the $z$-axis. Therefore, the kernel of $T$ is: $\ker(T) = \left\{ \begin{bmatrix} 0 \\ 0 \\ z \end{bmatrix} \mid z \in \mathbb{R} \right\} = \text{span} \left( \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right).$

Range of $T$:

The range (image) of $T$, denoted by $\text{Im}(T)$, consists of all vectors in $\mathbb{R}^3$ that can be written as $T(x, y, z)$. Since $T(x, y, z) = (x, y, 0)$, the range is the set of all vectors of the form $(x, y, 0)$, which is the $xy$-plane. Therefore, the range of $T$ is: $\text{Im}(T) = \left\{ \begin{bmatrix} x \\ y \\ 0 \end{bmatrix} \mid x, y \in \mathbb{R} \right\} = \text{span} \left( \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right).$

Summary:

• Kernel: The kernel of $T$ is the $z$-axis, i.e., $\ker(T) = \text{span} \left( \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right)$.
• Range: The range of $T$ is the $xy$-plane, i.e., $\text{Im}(T) = \text{span} \left( \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right)$.

Would you like more details or have any further questions?

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Here are 5 related questions for further exploration:

1. How do you compute the rank of a linear transformation?
2. What is the geometric interpretation of the kernel and range of a transformation?
3. Can a linear transformation from $\mathbb{R}^3 \to \mathbb{R}^3$ have both a trivial kernel and a trivial range?
4. How does the rank-nullity theorem apply to this transformation?
5. How can you find the inverse of a linear transformation if it exists?

Tip: The rank-nullity theorem is a powerful tool that relates the dimension of the kernel and range of a linear transformation to the dimension of the domain.