6. General Rank Nullity states: If T:VW is a linear transformation and V is finite-dimensional, then dim[Ker(T)]+dim[Rng(T)]=dim[V]. Use this information to answer the following questions: Suppose that the Kernel of T was a 2-dimensional subspace of RS and W is R³. a. What is V? b. Is T onto, explain how you know? C. Is T one-to-one? Explain how you know.
6. General Rank Nullity states: If T:VW is a linear transformation and V is finite-dimensional, then dim[Ker(T)]+dim[Rng(T)]=dim[V]. Use this information to answer the following questions: Suppose that the Kernel of T was a 2-dimensional subspace of RS and W is R³. a. What is V? b. Is T onto, explain how you know? C. Is T one-to-one? Explain how you know.
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 #6, Show all of your steps and all of your work please. Post your work on pictures please!
![### Linear Transformations and Applications
#### 3. Linear Transformation with Matrix
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. Suppose that \( T: \mathbb{R}^n \to \mathbb{R}^m \). What is \( n \) and what is \( m \)?
b. Find \( \text{Ker} (T) \).
c. Find \( \text{Rng} (T) \).
d. Determine if \( T \) is one-to-one.
e. Determine if \( T \) is onto.
#### 4. Integration and Orthogonality
Suppose that \( V = C^0[0, 1] \), and let \( (f(x), g(x)) = \int_0^1 f(x)g(x) \, dx \). You may use a calculator to integrate. Show what you put into the calculator as well as the results.
a. Find \( \langle x, 2x^3 \rangle \).
b. Find \( \| 3x \| \).
c. Determine if \( f(x) = \cos (\pi x) \) and \( g(x) = \sin(\pi x) \) is orthogonal. Explain your reasoning.
#### 5. Projections and Distance
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
The theorem states:
If \( T: V \to W \) is a linear transformation and \( V \) is finite-dimensional, then:
\[ \dim(\text{Ker}(T)) + \dim(\text{Rng}(T)) = \dim(V) \]
Use this information to answer the following questions:
Suppose that the Kernel of \( T \) was a 2-dimensional subspace of \( \mathbb{R}^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655393b-8df3-4633-b13c-e0d6983d2306%2F44198868-ac39-4606-a184-e7bc2ccc98e8%2Fhzo50vb_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformations and Applications
#### 3. Linear Transformation with Matrix
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. Suppose that \( T: \mathbb{R}^n \to \mathbb{R}^m \). What is \( n \) and what is \( m \)?
b. Find \( \text{Ker} (T) \).
c. Find \( \text{Rng} (T) \).
d. Determine if \( T \) is one-to-one.
e. Determine if \( T \) is onto.
#### 4. Integration and Orthogonality
Suppose that \( V = C^0[0, 1] \), and let \( (f(x), g(x)) = \int_0^1 f(x)g(x) \, dx \). You may use a calculator to integrate. Show what you put into the calculator as well as the results.
a. Find \( \langle x, 2x^3 \rangle \).
b. Find \( \| 3x \| \).
c. Determine if \( f(x) = \cos (\pi x) \) and \( g(x) = \sin(\pi x) \) is orthogonal. Explain your reasoning.
#### 5. Projections and Distance
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
The theorem states:
If \( T: V \to W \) is a linear transformation and \( V \) is finite-dimensional, then:
\[ \dim(\text{Ker}(T)) + \dim(\text{Rng}(T)) = \dim(V) \]
Use this information to answer the following questions:
Suppose that the Kernel of \( T \) was a 2-dimensional subspace of \( \mathbb{R}^
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