8. Let H be the set of all vectors of the forms given below and either find a basis for the vector space, or give an example to show that it is not a vector space. 39 – 4p 2p 2c b За — 26 - (a) : P, q are real (b) a, b, c are real q +1 [2р + 59 + 4b + 3c
8. Let H be the set of all vectors of the forms given below and either find a basis for the vector space, or give an example to show that it is not a vector space. 39 – 4p 2p 2c b За — 26 - (a) : P, q are real (b) a, b, c are real q +1 [2р + 59 + 4b + 3c
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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![### Problem Statement
Consider the set \( H \) defined as follows:
**8. Let \( H \) be the set of all vectors of the forms given below and either find a basis for the vector space, or give an example to show that it is not a vector space.**
#### (a) Vectors of the form
\[
\left\{ \begin{bmatrix} 3q - 4p \\ 2p \\ q + 1 \\ 2p + 5q \end{bmatrix} : p, q \text{ are real} \right\}
\]
#### (b) Vectors of the form
\[
\left\{ \begin{bmatrix} 2c - b \\ 3a - 2b \\ 0 \\ a + 4b + 3c \end{bmatrix} : a, b, c \text{ are real} \right\}
\]
#### Instructions
For each set of vectors, determine if the set forms a vector space. If it does, find a basis for the vector space. If it does not, provide an example to show why it is not a vector space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa51cb2c6-68d7-460b-b6c2-8a3d6ada438a%2F83e8249a-cc6a-4d30-ac1a-2bd9b5f78df5%2Ffcrt4t9_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Consider the set \( H \) defined as follows:
**8. Let \( H \) be the set of all vectors of the forms given below and either find a basis for the vector space, or give an example to show that it is not a vector space.**
#### (a) Vectors of the form
\[
\left\{ \begin{bmatrix} 3q - 4p \\ 2p \\ q + 1 \\ 2p + 5q \end{bmatrix} : p, q \text{ are real} \right\}
\]
#### (b) Vectors of the form
\[
\left\{ \begin{bmatrix} 2c - b \\ 3a - 2b \\ 0 \\ a + 4b + 3c \end{bmatrix} : a, b, c \text{ are real} \right\}
\]
#### Instructions
For each set of vectors, determine if the set forms a vector space. If it does, find a basis for the vector space. If it does not, provide an example to show why it is not a vector space.
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