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1-4 3. Let A = 3 0 and define T(x) = T₁(x) = Ax. [1 2 (a) Find T(x), the image of x = [4] under the transformation T. 3 (b) Determine if c = [] is in the range of the transformation T. (c) Find an æ in R² whose image under T is b = [ ³3 ] Is there more than one a whose image under T is b = [ [&] ? (d) Find all x in R2 that are mapped into the zero vector by the tranformation T.

1-4 3. Let A = 3 0 and define T(x) = T₁(x) = Ax. [1 2 (a) Find T(x), the image of x = [4] under the transformation T. 3 (b) Determine if c = [] is in the range of the transformation T. (c) Find an æ in R² whose image under T is b = [ ³3 ] Is there more than one a whose image under T is b = [ [&] ? (d) Find all x in R2 that are mapped into the zero vector by the tranformation T.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 26E
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### Linear Algebra Transformation Problem #### Problem Statement Let \( A = \begin{bmatrix} 1 & -4 \\ 3 & 0 \\ 1 & 2 \end{bmatrix} \) and define \( T(x) = T_A(x) = Ax \). 1. **(a)** Find \( T(x) \), the image of \( x = \begin{bmatrix} 4 \\ -1 \end{bmatrix} \) under the transformation \( T \). 2. **(b)** Determine if \( c = \begin{bmatrix} 3 \\ 0 \\ -1 \end{bmatrix} \) is in the range of the transformation \( T \). 3. **(c)** Find an \( x \) in \( \mathbb{R}^2 \) whose image under \( T \) is \( b = \begin{bmatrix} 9 \\ 3 \\ -3 \end{bmatrix} \). - Is there more than one \( x \) whose image under \( T \) is \( b = \begin{bmatrix} 9 \\ 3 \\ -3 \end{bmatrix} \)? 4. **(d)** Find all \( x \) in \( \mathbb{R}^2 \) that are mapped into the zero vector by the transformation \( T \). #### Explanation In this problem, we are dealing with a linear transformation \( T \) defined by the matrix \( A \), which maps vectors from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \). * **Part (a):** You are asked to compute the image of a given vector \( x \) under the transformation \( T \). * **Part (b):** You need to check if a specific vector \( c \) is within the range of the transformation, meaning if there exists some \( x \in \mathbb{R}^2 \) such that \( T(x) = c \). * **Part (c):** Here, you're required to find a specific vector \( x \) such that its image under \( T \) matches a given vector \( b \). Additionally, you need to determine the uniqueness of such an \( x \). * **Part (d):** This part involves finding all vectors \( x \) that result in the zero
Transcribed Image Text:### Linear Algebra Transformation Problem #### Problem Statement Let \( A = \begin{bmatrix} 1 & -4 \\ 3 & 0 \\ 1 & 2 \end{bmatrix} \) and define \( T(x) = T_A(x) = Ax \). 1. **(a)** Find \( T(x) \), the image of \( x = \begin{bmatrix} 4 \\ -1 \end{bmatrix} \) under the transformation \( T \). 2. **(b)** Determine if \( c = \begin{bmatrix} 3 \\ 0 \\ -1 \end{bmatrix} \) is in the range of the transformation \( T \). 3. **(c)** Find an \( x \) in \( \mathbb{R}^2 \) whose image under \( T \) is \( b = \begin{bmatrix} 9 \\ 3 \\ -3 \end{bmatrix} \). - Is there more than one \( x \) whose image under \( T \) is \( b = \begin{bmatrix} 9 \\ 3 \\ -3 \end{bmatrix} \)? 4. **(d)** Find all \( x \) in \( \mathbb{R}^2 \) that are mapped into the zero vector by the transformation \( T \). #### Explanation In this problem, we are dealing with a linear transformation \( T \) defined by the matrix \( A \), which maps vectors from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \). * **Part (a):** You are asked to compute the image of a given vector \( x \) under the transformation \( T \). * **Part (b):** You need to check if a specific vector \( c \) is within the range of the transformation, meaning if there exists some \( x \in \mathbb{R}^2 \) such that \( T(x) = c \). * **Part (c):** Here, you're required to find a specific vector \( x \) such that its image under \( T \) matches a given vector \( b \). Additionally, you need to determine the uniqueness of such an \( x \). * **Part (d):** This part involves finding all vectors \( x \) that result in the zero
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