MATH 310 1. Let e₁,e2,e3 be the standard basis vectors in R³ and consider the ordered basis: [e3, -e1,e₂+ e1] Verify that this is actually a basis and find the coordinates of the vector (1,1,1) with respect to that basis. 2. Let T be the linear map from R² to R² defined by: T((z. y)²) = (x - 2y. 2x + y)² Find its matrix (with respect to the standard basis of R²). 3. Let T be the linear map from R³ to R³ defined by the formula: T((z. y. 2)²) = (3x + y, 2r-z, y)² Find the matrix of T with respect to the standard basis of R³.
MATH 310 1. Let e₁,e2,e3 be the standard basis vectors in R³ and consider the ordered basis: [e3, -e1,e₂+ e1] Verify that this is actually a basis and find the coordinates of the vector (1,1,1) with respect to that basis. 2. Let T be the linear map from R² to R² defined by: T((z. y)²) = (x - 2y. 2x + y)² Find its matrix (with respect to the standard basis of R²). 3. Let T be the linear map from R³ to R³ defined by the formula: T((z. y. 2)²) = (3x + y, 2r-z, y)² Find the matrix of T with respect to the standard basis of R³.
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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Question
![**MATH 310**
1. Let \( e_1, e_2, e_3 \) be the standard basis vectors in \( \mathbb{R}^3 \) and consider the ordered basis:
\[
\{e_3, -e_1, e_2 + e_1\}
\]
Verify that this is actually a basis and find the coordinates of the vector \( (1, 1, 1)^T \) with respect to that basis.
2. Let \( T \) be the linear map from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \) defined by:
\[
T((x, y)^T) = (x - 2y, 2x + y)^T
\]
Find its matrix (with respect to the standard basis of \( \mathbb{R}^2 \)).
3. Let \( T \) be the linear map from \( \mathbb{R}^3 \) to \( \mathbb{R}^3 \) defined by the formula:
\[
T((x, y, z)^T) = (3x + y, 2x - z, y)^T
\]
Find the matrix of \( T \) with respect to the standard basis of \( \mathbb{R}^3 \).
---
**Detailed Explanation of Concepts**
1. **Standard Basis in \( \mathbb{R}^3 \)**:
The standard basis of \( \mathbb{R}^3 \) consists of the vectors:
\[
e_1 = (1, 0, 0)^T, \quad e_2 = (0, 1, 0)^T, \quad e_3 = (0, 0, 1)^T
\]
To verify that \( \{e_3, -e_1, e_2 + e_1\} \) is a basis, we need to show that these vectors are linearly independent and span \( \mathbb{R}^3 \).
2. **Linear Map and Matrices**:
For a linear map \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda22bb93-89e0-43b0-b495-0a6067b20da3%2F5ab798b4-ba22-4441-82bf-3d7cf59b4b7e%2F3de4q8_processed.png&w=3840&q=75)
Transcribed Image Text:**MATH 310**
1. Let \( e_1, e_2, e_3 \) be the standard basis vectors in \( \mathbb{R}^3 \) and consider the ordered basis:
\[
\{e_3, -e_1, e_2 + e_1\}
\]
Verify that this is actually a basis and find the coordinates of the vector \( (1, 1, 1)^T \) with respect to that basis.
2. Let \( T \) be the linear map from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \) defined by:
\[
T((x, y)^T) = (x - 2y, 2x + y)^T
\]
Find its matrix (with respect to the standard basis of \( \mathbb{R}^2 \)).
3. Let \( T \) be the linear map from \( \mathbb{R}^3 \) to \( \mathbb{R}^3 \) defined by the formula:
\[
T((x, y, z)^T) = (3x + y, 2x - z, y)^T
\]
Find the matrix of \( T \) with respect to the standard basis of \( \mathbb{R}^3 \).
---
**Detailed Explanation of Concepts**
1. **Standard Basis in \( \mathbb{R}^3 \)**:
The standard basis of \( \mathbb{R}^3 \) consists of the vectors:
\[
e_1 = (1, 0, 0)^T, \quad e_2 = (0, 1, 0)^T, \quad e_3 = (0, 0, 1)^T
\]
To verify that \( \{e_3, -e_1, e_2 + e_1\} \) is a basis, we need to show that these vectors are linearly independent and span \( \mathbb{R}^3 \).
2. **Linear Map and Matrices**:
For a linear map \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \(
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