# Solution ## Step 1: Express the Linear Transformation in Matrix Form The transformation \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) is defined as: \[ T(x, y, z) = (x-y,\, y-z,\, x-z) \] This can be written in matrix form as: \[ T(x, y, z) = \begin{pmatrix} 1 & -1 & \\ & 1 & -1 \\ 1 & & -1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \] Let \( A \) denote the matrix of \( T \): \[ A = \begin{pmatrix} 1 & -1 & \\ & 1 & -1 \\ 1 & & -1 \end{pmatrix} \] ## Step 2: Find the Image of \( T \) (Column Space of \( A \)) To describe the image, consider a general vector \( (a, b, c) \) in the image: \[ (a, b, c) = (x-y,\, y-z,\, x-z) \] Express each variable in terms of \( x, y, z \): - \( a = x - y \) - \( b = y - z \) - \( c = x - z \) Notice that: \[ c = (x - z) = (x - y) + (y - z) = a + b \] Therefore, every vector in the image satisfies \( c = a + b \). ## Step 3: Explicit Description of the Image The image of \( T \) consists of all vectors \( (a, b, c) \in \mathbb{R}^3 \) such that \( c = a + b \). This can be written as: \[ \text{Im}(T) = \{ (a, b, a+b) \mid a, b \in \mathbb{R} \} \] This set is a plane in \( \mathbb{R}^3 \). **Interpretation:** The image is a 2-dimensional subspace of \( \mathbb{R}^3 \) where the third component is always the sum of the first two. ## Step 4: Find the Dimension of the Image Choose basis vectors for the image: - For \( a = 1, b = \): \( (1, , 1) \) - For \( a = , b = 1 \): \( (, 1, 1) \) These two vectors are linearly independent and span the image. Thus, the dimension is 2. ## Step 5: Final Answer **Explicit description:** The image of \( T \) is the set of all vectors \( (a, b, a+b) \) where \( a, b \in \mathbb{R} \). **Dimension:** The dimension of the image of \( T \) is **2**. --- **Summary:** The image of \( T \) is the plane \(\{(a, b, a+b) : a, b \in \mathbb{R}\}\) in \(\mathbb{R}^3\), and its dimension is 2.