Let T be a linear transformation from R² to R³ and suppose that (Q)-[]). T ¹ ([³]) - the vector [-2.] 5 3 and T as a linear combination of vectors [3] and [A] Show your
Let T be a linear transformation from R² to R³ and suppose that (Q)-[]). T ¹ ([³]) - the vector [-2.] 5 3 and T as a linear combination of vectors [3] and [A] Show your
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
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,...
Related questions
Question
![**Problem Statement:**
Let \( T \) be a linear transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \) and suppose that
\[
T \left( \begin{bmatrix} 2 \\ 3 \end{bmatrix} \right) = \begin{bmatrix} 5 \\ 3 \\ -1 \end{bmatrix}, \quad \text{and} \quad T \left( \begin{bmatrix} 5 \\ -1 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}.
\]
**Task (a):**
Express the vector \( \begin{bmatrix} 2 \\ -14 \end{bmatrix} \) as a linear combination of vectors \( \begin{bmatrix} 2 \\ 3 \end{bmatrix} \) and \( \begin{bmatrix} 5 \\ -1 \end{bmatrix} \). Show your work.
**Task (b):**
Use the properties of linear transformations and your answer to part (a) to find \( T \left( \begin{bmatrix} 2 \\ -14 \end{bmatrix} \right) \).
---
**Visual Representation:**
There are no graphs or diagrams included in the problem statement. The problem consists of matrices and expressions related to linear transformations, which are presented in matrix form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd4f4b624-9180-41b5-94fa-c093ec7455d7%2F9daffed7-79cb-4662-a267-a5221235c1a1%2F5kg1e2t_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( T \) be a linear transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \) and suppose that
\[
T \left( \begin{bmatrix} 2 \\ 3 \end{bmatrix} \right) = \begin{bmatrix} 5 \\ 3 \\ -1 \end{bmatrix}, \quad \text{and} \quad T \left( \begin{bmatrix} 5 \\ -1 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}.
\]
**Task (a):**
Express the vector \( \begin{bmatrix} 2 \\ -14 \end{bmatrix} \) as a linear combination of vectors \( \begin{bmatrix} 2 \\ 3 \end{bmatrix} \) and \( \begin{bmatrix} 5 \\ -1 \end{bmatrix} \). Show your work.
**Task (b):**
Use the properties of linear transformations and your answer to part (a) to find \( T \left( \begin{bmatrix} 2 \\ -14 \end{bmatrix} \right) \).
---
**Visual Representation:**
There are no graphs or diagrams included in the problem statement. The problem consists of matrices and expressions related to linear transformations, which are presented in matrix form.
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