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Find a basis for the kernel of the linear map T from R³ to R² given by: T((x, y, z)) = (x + y, z)

Find a basis for the kernel of the linear map T from R³ to R² given by: T((x, y, z)) = (x + y, z)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.8: Determinants
Problem 8E
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### Linear Algebra: Finding Bases for Kernels and Images of Linear Maps **Problem 2: Finding a Basis for the Kernel** - **Task**: Find a basis for the kernel of the linear map \( T \) from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \) given by: \[ T((x, y, z)^T) = (x + y, z)^T \] **Solution Approach**: - Given the transformation \( T \), we are to determine the set of all vectors \( (x, y, z) \) in \( \mathbb{R}^3 \) that are mapped to the zero vector in \( \mathbb{R}^2 \). - This implies solving the equation \( T((x, y, z)^T) = (0, 0)^T \). **Steps**: 1. Set up the equations based on \( T \): \[ x + y = 0 \] \[ z = 0 \] 2. Solve the equations: - From \( x + y = 0 \), we get \( y = -x \). - From \( z = 0 \), we simply have \( z = 0 \). 3. Express the solution in vector form: \[ (x, y, z) = (x, -x, 0) = x(1, -1, 0) \] 4. Conclude that a basis for the kernel (set of vectors) is \( \{(1, -1, 0)\} \). **Final Answer**: \[ \{(1, -1, 0)\} \] **Problem 3: Finding a Basis for the Image** - **Task**: Find a basis for the image of the linear map \( T \) from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \) given by: \[ T((x, y)^T) = (x, x + y, y)^T \] **Solution Approach**: - Given the transformation \( T \), we are to determine the set of all vectors that can be written in the form \( (x, x + y, y) \) in \( \mathbb{R}^3 \). **Steps**: 1. Understand that \( T \)
Transcribed Image Text:### Linear Algebra: Finding Bases for Kernels and Images of Linear Maps **Problem 2: Finding a Basis for the Kernel** - **Task**: Find a basis for the kernel of the linear map \( T \) from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \) given by: \[ T((x, y, z)^T) = (x + y, z)^T \] **Solution Approach**: - Given the transformation \( T \), we are to determine the set of all vectors \( (x, y, z) \) in \( \mathbb{R}^3 \) that are mapped to the zero vector in \( \mathbb{R}^2 \). - This implies solving the equation \( T((x, y, z)^T) = (0, 0)^T \). **Steps**: 1. Set up the equations based on \( T \): \[ x + y = 0 \] \[ z = 0 \] 2. Solve the equations: - From \( x + y = 0 \), we get \( y = -x \). - From \( z = 0 \), we simply have \( z = 0 \). 3. Express the solution in vector form: \[ (x, y, z) = (x, -x, 0) = x(1, -1, 0) \] 4. Conclude that a basis for the kernel (set of vectors) is \( \{(1, -1, 0)\} \). **Final Answer**: \[ \{(1, -1, 0)\} \] **Problem 3: Finding a Basis for the Image** - **Task**: Find a basis for the image of the linear map \( T \) from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \) given by: \[ T((x, y)^T) = (x, x + y, y)^T \] **Solution Approach**: - Given the transformation \( T \), we are to determine the set of all vectors that can be written in the form \( (x, x + y, y) \) in \( \mathbb{R}^3 \). **Steps**: 1. Understand that \( T \)
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