%3D [3 2. If T is a linear transformation with T: R? - R? and T 3 and T 2. find the 2. 11 standard matrix for the linear transformation T.
%3D [3 2. If T is a linear transformation with T: R? - R? and T 3 and T 2. find the 2. 11 standard matrix for the linear transformation T.
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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![**Linear Algebra Homework Questions**
---
**Instructions:**
- Carefully read each question and provide detailed responses.
- Use proper mathematical notation and set theories.
- Show all steps in your workings for full marks.
---
1. **Problem 1:**
If \( T \) is a linear transformation with \( T:\mathbb{R}^2 \to \mathbb{R}^2 \) and
\[
T \left[ \begin{array}{c} 1 \\ 2 \end{array} \right] = \left[ \begin{array}{c} 5 \\ 2 \end{array} \right] \quad \text{and} \quad T \left[ \begin{array}{c} -3 \\ 1 \end{array} \right] = \left[ \begin{array}{c} 3 \\ 1 \end{array} \right],
\]
find the standard matrix for the linear transformation \( T \).
2. **Problem 2:**
Give the solution set of the equation \( x - 2y + 3z = 12 \) using proper set notation.
3. **Problem 3:**
Are the vectors
\[
\left[ \begin{array}{c} 1 \\ 3 \\ 5 \end{array} \right], \quad \left[ \begin{array}{c} 2 \\ 3 \\ 1 \end{array} \right], \quad \left[ \begin{array}{c} 2 \\ 5 \\ 2 \end{array} \right]
\]
linearly independent? If not, then write a dependency relation.
4. **Problem 4:**
\( T \) is a linear transformation with \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) defined by
\[
T \left( \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right] \right) = \left[ \begin{array}{c} x_1 + x_2 - x_3 \\ x_1 - x_2 + x_3 \\ x_1 + x_3 \end{array} \right].
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2Fb24e6c13-7924-4478-ad49-5c5669f640ce%2Fxparw2_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Homework Questions**
---
**Instructions:**
- Carefully read each question and provide detailed responses.
- Use proper mathematical notation and set theories.
- Show all steps in your workings for full marks.
---
1. **Problem 1:**
If \( T \) is a linear transformation with \( T:\mathbb{R}^2 \to \mathbb{R}^2 \) and
\[
T \left[ \begin{array}{c} 1 \\ 2 \end{array} \right] = \left[ \begin{array}{c} 5 \\ 2 \end{array} \right] \quad \text{and} \quad T \left[ \begin{array}{c} -3 \\ 1 \end{array} \right] = \left[ \begin{array}{c} 3 \\ 1 \end{array} \right],
\]
find the standard matrix for the linear transformation \( T \).
2. **Problem 2:**
Give the solution set of the equation \( x - 2y + 3z = 12 \) using proper set notation.
3. **Problem 3:**
Are the vectors
\[
\left[ \begin{array}{c} 1 \\ 3 \\ 5 \end{array} \right], \quad \left[ \begin{array}{c} 2 \\ 3 \\ 1 \end{array} \right], \quad \left[ \begin{array}{c} 2 \\ 5 \\ 2 \end{array} \right]
\]
linearly independent? If not, then write a dependency relation.
4. **Problem 4:**
\( T \) is a linear transformation with \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) defined by
\[
T \left( \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right] \right) = \left[ \begin{array}{c} x_1 + x_2 - x_3 \\ x_1 - x_2 + x_3 \\ x_1 + x_3 \end{array} \right].
\]
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