([₁]) = Justify your answer. If T is a linear transformation standard basis of R². 7.1. Let T R2 R² be defined by T →>> x .2 Is T a linear transformation? find its matrix relative to the
([₁]) = Justify your answer. If T is a linear transformation standard basis of R². 7.1. Let T R2 R² be defined by T →>> x .2 Is T a linear transformation? find its matrix relative to the
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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I'm currently encountering difficulties in solving this problem using matrix notation alone, and I'm looking for your guidance. The problem specifically requires a solution using matrix notation exclusively, without employing any other methods. Can you please provide a thorough, step-by-step explanation in matrix notation to assist me in reaching the final solution?
matrix way only
I have attached the question and answer can you use the matrix way leading up to the solution
![### Linear Transformation Example
#### 7.1
Consider the linear transformation \( T \):
\[
T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}
\]
However, if we apply the transformation to the scalar multiple of the vector:
\[
T \left( 2 \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = T \left( \begin{bmatrix} 2 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 2 \\ 4 \end{bmatrix}
\]
We observe that:
\[
\begin{bmatrix} 2 \\ 4 \end{bmatrix} \neq 2T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right)
\]
This indicates that \( T \) is not a linear transformation since it does not satisfy the property of linearity, specifically the principle of scalar multiplication \( T(c\mathbf{v}) = cT(\mathbf{v}) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F207ac185-b7c0-499b-9b4e-65755801eeb3%2Fa9c31c10-d8ed-448a-a448-1c2daa60f167%2Frcbnyh_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformation Example
#### 7.1
Consider the linear transformation \( T \):
\[
T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}
\]
However, if we apply the transformation to the scalar multiple of the vector:
\[
T \left( 2 \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = T \left( \begin{bmatrix} 2 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 2 \\ 4 \end{bmatrix}
\]
We observe that:
\[
\begin{bmatrix} 2 \\ 4 \end{bmatrix} \neq 2T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right)
\]
This indicates that \( T \) is not a linear transformation since it does not satisfy the property of linearity, specifically the principle of scalar multiplication \( T(c\mathbf{v}) = cT(\mathbf{v}) \).
![### Problem 7.1: Determining Linearity of a Transformation
Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be defined by
\[
T \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} x \\ x^2 \end{bmatrix}.
\]
Is \( T \) a linear transformation? Justify your answer. If \( T \) is a linear transformation, find its matrix relative to the standard basis of \( \mathbb{R}^2 \).
### Explanation:
1. **Determine Linearity:**
To check if \( T \) is a linear transformation, we need to verify if it satisfies the following two properties for all vectors \( \mathbf{u}, \mathbf{v} \in \mathbb{R}^2 \) and all scalars \( c \in \mathbb{R} \):
- **Additivity:** \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).
- **Homogeneity:** \( T(c \mathbf{u}) = c T(\mathbf{u}) \).
2. **Matrix Representation of \( T \):**
If \( T \) is determined to be a linear transformation, we would find a matrix \( A \) such that for any vector \( \mathbf{v} \in \mathbb{R}^2 \),
\[
T(\mathbf{v}) = A \mathbf{v}
\]
where \( \mathbf{v} \) is expressed in terms of the standard basis of \( \mathbb{R}^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F207ac185-b7c0-499b-9b4e-65755801eeb3%2Fa9c31c10-d8ed-448a-a448-1c2daa60f167%2F23qf82k_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 7.1: Determining Linearity of a Transformation
Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be defined by
\[
T \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} x \\ x^2 \end{bmatrix}.
\]
Is \( T \) a linear transformation? Justify your answer. If \( T \) is a linear transformation, find its matrix relative to the standard basis of \( \mathbb{R}^2 \).
### Explanation:
1. **Determine Linearity:**
To check if \( T \) is a linear transformation, we need to verify if it satisfies the following two properties for all vectors \( \mathbf{u}, \mathbf{v} \in \mathbb{R}^2 \) and all scalars \( c \in \mathbb{R} \):
- **Additivity:** \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).
- **Homogeneity:** \( T(c \mathbf{u}) = c T(\mathbf{u}) \).
2. **Matrix Representation of \( T \):**
If \( T \) is determined to be a linear transformation, we would find a matrix \( A \) such that for any vector \( \mathbf{v} \in \mathbb{R}^2 \),
\[
T(\mathbf{v}) = A \mathbf{v}
\]
where \( \mathbf{v} \) is expressed in terms of the standard basis of \( \mathbb{R}^2 \).
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