## Problem Restatement Given the linear transformation \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) defined by: \[ T(x, y, z) = (x - y, y - z, x - z) \] Describe the image of \( T \) explicitly and determine its dimension. --- ## Step 1: Write the Output as a Vector Let \((a, b, c) = (x - y, y - z, x - z)\). --- ## Step 2: Find Relationships From the transformation: - \( a = x - y \) - \( b = y - z \) - \( c = x - z \) Notice that: \[ a + b = (x - y) + (y - z) = x - z = c \] So every vector in the image satisfies \( a + b = c \). --- ## Step 3: Explicit Description of the Image Thus, the image of \( T \) is: \[ \text{Im}(T) = \left\{ (a, b, c) \in \mathbb{R}^3 \mid c = a + b \right\} \] This is a plane in \(\mathbb{R}^3\). --- ## Step 4: Dimension of the Image The equation \( c = a + b \) defines a plane, which is **2-dimensional**. --- ## Step 5: Basis for the Image (Optional) To find a basis, set \( a \) and \( b \) as free variables: \[ (a, b, a + b) = a(1, , 1) + b(, 1, 1) \] So a basis is \( \{ (1, , 1), (, 1, 1) \} \). --- ## Final Answers ### **Explicit Description** \[ \boxed{ \text{Im}(T) = \left\{ (a, b, c) \in \mathbb{R}^3 \mid c = a + b \right\} } \] ### **Dimension** \[ \boxed{2} \]

# Verification of the Solution ## Problem Summary We are given a linear transformation \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) defined by: \[ T(x, y, z) = (x - y, y - z, x - z) \] and need to describe the image of \( T \) and determine its dimension. ## Step-by-Step Analysis ### Step 1: Define the Output Let \((a, b, c) = (x - y, y - z, x - z)\). ### Step 2: Establish Relationships From the transformation, we have: - \( a = x - y \) - \( b = y - z \) - \( c = x - z \) We can derive the following relationship: \[ a + b = c \] This indicates that any vector in the image of \( T \) must satisfy this equation. ### Step 3: Explicit Description of the Image The image of \( T \) can thus be expressed as: \[ \text{Im}(T) = \left\{ (a, b, c) \in \mathbb{R}^3 \mid c = a + b \right\} \] This represents a plane in \(\mathbb{R}^3\). ### Step 4: Determine the Dimension The equation \( c = a + b \) describes a plane, which is 2-dimensional. ### Step 5: Basis for the Image To find a basis for this image, we can select \( a \) and \( b \) as free variables: - Let \( a = 1 \) and \( b = 0 \) gives the vector \( (1, 0, 1) \). - Let \( a = 0 \) and \( b = 1 \) gives the vector \( (0, 1, 1) \). Thus, a basis for the image is: \[ \{ (1, 0, 1), (0, 1, 1) \} \] ## Final Results ### Explicit Description of the Image \[ \boxed{ \text{Im}(T) = \left\{ (a, b, c) \in \mathbb{R}^3 \mid c = a + b \right\} } \] ### Dimension of the Image \[ \boxed{2} \] ## Conclusion The initial analysis and conclusions are correct. The image of the transformation \( T \) is a plane in \(\mathbb{R}^3\) with a dimension of 2.