5. 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.
5. 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.
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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Question
SOlve #5, Show all of your steps and all of your work please. Post your work on pictures please!
![**Linear Algebra Problem Set**
**3.** 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 \( T: \mathbb{R}^n \rightarrow \mathbb{R}^m \). 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.** 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.** 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:
If \( T: V \rightarrow W \) 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 \( \mathbb{R}^5 \) and \( W \) is \( \mathbb{R}^3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655393b-8df3-4633-b13c-e0d6983d2306%2F13ae3a9c-c46e-40e6-95a7-cc08d5f01d3d%2Fwv3ijcb_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Problem Set**
**3.** 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 \( T: \mathbb{R}^n \rightarrow \mathbb{R}^m \). 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.** 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.** 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:
If \( T: V \rightarrow W \) 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 \( \mathbb{R}^5 \) and \( W \) is \( \mathbb{R}^3
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