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)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 \( T \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda22bb93-89e0-43b0-b495-0a6067b20da3%2F2ab83f88-39bd-4575-af64-19d7dc44da0c%2Fvazc424_processed.png&w=3840&q=75)
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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