2 Let T: R2 R2 be a linear transformation that maps u = 4 The image of 2u is. into 5 1 and maps v= 4 into -1 4 Use the fact that T is linear to find the images under T of 2u, 3v, and 2u + 3v.
2 Let T: R2 R2 be a linear transformation that maps u = 4 The image of 2u is. into 5 1 and maps v= 4 into -1 4 Use the fact that T is linear to find the images under T of 2u, 3v, and 2u + 3v.
Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
1.8 #5
![**Linear Transformation Problem**
Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation that maps:
\[
u = \begin{bmatrix} 2 \\ 4 \end{bmatrix}
\]
into
\[
\begin{bmatrix} 5 \\ 1 \end{bmatrix}
\]
and maps:
\[
v = \begin{bmatrix} 4 \\ 4 \end{bmatrix}
\]
into
\[
\begin{bmatrix} 4 \\ 4 \end{bmatrix}
\]
*Use the fact that \( T \) is linear to find the images under \( T \) of \( 2u \), \( 3v \), and \( 2u + 3v \).*
**Calculations:**
1. **The Image of \( 2u \)**:
- Since \( T \) is linear, \( T(2u) = 2T(u) \).
- Therefore, the image of \( 2u \) is \( 2 \times \begin{bmatrix} 5 \\ 1 \end{bmatrix} = \begin{bmatrix} 10 \\ 2 \end{bmatrix} \).
**Question Viewer and Answer Box:**
- At the bottom, there is an interface with various mathematical symbols, which users can use to input their answers.
- The text box reads: "The image of \( 2u \) is \(\_\_\_\)."
**Note:** The steps for finding the images of \( 3v \) and \( 2u + 3v \) would follow a similar linearity principle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ec5dee8-4b81-4c1e-b6ca-023c2626018b%2F166314de-3835-4923-82cb-97c70b239e73%2F8hdqjzv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Transformation Problem**
Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation that maps:
\[
u = \begin{bmatrix} 2 \\ 4 \end{bmatrix}
\]
into
\[
\begin{bmatrix} 5 \\ 1 \end{bmatrix}
\]
and maps:
\[
v = \begin{bmatrix} 4 \\ 4 \end{bmatrix}
\]
into
\[
\begin{bmatrix} 4 \\ 4 \end{bmatrix}
\]
*Use the fact that \( T \) is linear to find the images under \( T \) of \( 2u \), \( 3v \), and \( 2u + 3v \).*
**Calculations:**
1. **The Image of \( 2u \)**:
- Since \( T \) is linear, \( T(2u) = 2T(u) \).
- Therefore, the image of \( 2u \) is \( 2 \times \begin{bmatrix} 5 \\ 1 \end{bmatrix} = \begin{bmatrix} 10 \\ 2 \end{bmatrix} \).
**Question Viewer and Answer Box:**
- At the bottom, there is an interface with various mathematical symbols, which users can use to input their answers.
- The text box reads: "The image of \( 2u \) is \(\_\_\_\)."
**Note:** The steps for finding the images of \( 3v \) and \( 2u + 3v \) would follow a similar linearity principle.
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