Provide an example of a linear transformation T : R
3 →
R
3
that has a non-trivial kernel. Describe the kernel and range of this
transformation.


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.

Provide an example of a linear transformation T : R^3 → R^3 that has a non-trivial kernel. Describe the kernel and range of this transformation.

Explore a linear transformation from R^3 to R^3 with a non-trivial kernel, along with a step-by-step breakdown of the kernel and range. Learn about projections onto the xy-plane and the application of the rank-nullity theorem in linear algebra.

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