4. Suppose that V = Cº[0, 1], and let (f(x), g(x)) = f(x) g(x) dx. You may use a calculator to integrate. Show what you put into the calculator as well as the results. a. Find (x, 2x³). b. Find ||3x|| c. Determine if f(x) = cos (x) and g(x) = sin(x) is orthogonal. Explain your reasoning.
4. Suppose that V = Cº[0, 1], and let (f(x), g(x)) = f(x) g(x) dx. You may use a calculator to integrate. Show what you put into the calculator as well as the results. a. Find (x, 2x³). b. Find ||3x|| c. Determine if f(x) = cos (x) and g(x) = sin(x) is orthogonal. Explain your reasoning.
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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SOlve #4, Show all of your steps and all of your work please. Post your work on pictures please!
![### Linear Algebra Problem Set
#### 3. Linear Transformation Analysis
Suppose that \( T \) is a linear transformation with the matrix of the transformation:
\[ A = \begin{bmatrix} 1 & -2 & 5 \\ 2 & 3 & -4 \\ 1 & -5 & 11 \\ 0 & -2 & 4 \end{bmatrix} \]
- **a. Transformation Dimensions**
What are \( n \) and \( m \)?
- **b. Kernel of \( T \)**
Find \( \text{Ker} (T) \).
- **c. Range of \( T \)**
Find \( \text{Rng}(T) \).
- **d. One-to-One Property**
Determine if \( T \) is one-to-one.
- **e. Onto Property**
Determine if \( T \) is onto.
#### 4. Function Space Analysis
Suppose that \( V = C^0 [0, 1] \), and let \( (f(x), g(x)) = \int_0^1 f(x)g(x) \, dx \).
- **a. Polynomial Functions**
- Find \( \langle x, 2x^3 \rangle \).
- **b. Norm Evaluation**
- Find \( ||3x|| \).
- **c. Orthogonality Check**
- Determine if \( f(x) = \cos(\pi x) \) and \( g(x) = \sin(\pi x) \) are orthogonal. Explain your reasoning.
#### 5. Geometry and Distance Calculation
Use projections to find the distance from the point \((0, -1, 3)\) to the plane \(2x - 3y + z = 4\). Show your work. Write your answer as an exact answer and not as a decimal. You may use a calculator.
#### 6. General Rank Nullity Theorem
General Rank Nullity states:
If \( T: V \rightarrow W \) is a linear transformation and \( V \) is finite-dimensional, then:
\[ \text{dim}[\text{Ker}(T)] + \text{dim}[\text{Rng}(T)] = \text{dim}[V] \]
Use this information to answer the following questions:
Suppose that the Kernel](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655393b-8df3-4633-b13c-e0d6983d2306%2F1e40f59b-1354-44e7-a5d1-94119d656de6%2Fq2lrrd_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem Set
#### 3. Linear Transformation Analysis
Suppose that \( T \) is a linear transformation with the matrix of the transformation:
\[ A = \begin{bmatrix} 1 & -2 & 5 \\ 2 & 3 & -4 \\ 1 & -5 & 11 \\ 0 & -2 & 4 \end{bmatrix} \]
- **a. Transformation Dimensions**
What are \( n \) and \( m \)?
- **b. Kernel of \( T \)**
Find \( \text{Ker} (T) \).
- **c. Range of \( T \)**
Find \( \text{Rng}(T) \).
- **d. One-to-One Property**
Determine if \( T \) is one-to-one.
- **e. Onto Property**
Determine if \( T \) is onto.
#### 4. Function Space Analysis
Suppose that \( V = C^0 [0, 1] \), and let \( (f(x), g(x)) = \int_0^1 f(x)g(x) \, dx \).
- **a. Polynomial Functions**
- Find \( \langle x, 2x^3 \rangle \).
- **b. Norm Evaluation**
- Find \( ||3x|| \).
- **c. Orthogonality Check**
- Determine if \( f(x) = \cos(\pi x) \) and \( g(x) = \sin(\pi x) \) are orthogonal. Explain your reasoning.
#### 5. Geometry and Distance Calculation
Use projections to find the distance from the point \((0, -1, 3)\) to the plane \(2x - 3y + z = 4\). Show your work. Write your answer as an exact answer and not as a decimal. You may use a calculator.
#### 6. General Rank Nullity Theorem
General Rank Nullity states:
If \( T: V \rightarrow W \) is a linear transformation and \( V \) is finite-dimensional, then:
\[ \text{dim}[\text{Ker}(T)] + \text{dim}[\text{Rng}(T)] = \text{dim}[V] \]
Use this information to answer the following questions:
Suppose that the Kernel
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