For the subspace below, find a basis 9a + 18b3c 3a-b-c -6a +5b+2c -3a+b+c a, b, c in R a. (9,3, -6,3), (18,-1,-5,1) (6,3, -6, -3), (8,-1,5, -1) (6,3, -5,-3), (18,-1,5,1) (9,3, -6,3), (9,- 1,5,1) (9,3, -6, -3), (18,-1,5,1) C. d. e.
For the subspace below, find a basis 9a + 18b3c 3a-b-c -6a +5b+2c -3a+b+c a, b, c in R a. (9,3, -6,3), (18,-1,-5,1) (6,3, -6, -3), (8,-1,5, -1) (6,3, -5,-3), (18,-1,5,1) (9,3, -6,3), (9,- 1,5,1) (9,3, -6, -3), (18,-1,5,1) C. d. e.
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
![**Problem Statement: Finding a Basis for a Subspace**
Consider the following subspace of \(\mathbb{R}^4\):
\[
\left\{
\begin{pmatrix}
9a + 18b - 3c \\
3a - b - c \\
-6a + 5b + 2c \\
-3a + b + c
\end{pmatrix}
: a, b, c \in \mathbb{R}
\right\}
\]
To determine a basis for this subspace, we are presented with the following options:
**Options:**
a. \(\{(9, 3, -6, 3), (18, -1, -5, 1)\}\)
b. \(\{(6, 3, -6, -3), (8, -1.5, -1)\}\)
c. \(\{(6, 3, -5, -3), (18, -1, 1.5, 1)\}\)
d. \(\{(9, 3, -6, 3), (9, -1, 5, 1)\}\)
e. \(\{(9, 3, -6, -3), (18, -1, 5, 1)\}\)
**Objective:**
Identify the correct pair of vectors that form a basis for the given subspace. To form a basis, the vectors must be linearly independent and span the subspace.
**Analysis:**
- The vector pairs provided represent sets of vectors in \(\mathbb{R}^4\).
- Evaluate linear independence by ensuring that no vector in the set can be written as a linear combination of the other vectors in the set.
- Ensure that the vectors span the subspace by ensuring that every vector in this subspace can be written as a linear combination of the vectors in the selected pair.
This problem invites students to apply their understanding of linear algebra concepts, particularly subspaces, bases, and linear independence, to find the appropriate set of vectors that define the basis for the given subspace.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5483afab-7850-40f6-959f-e698dea1419a%2F427c0b9a-19d1-4bac-83b3-093d39e5009c%2Fde79qu_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Finding a Basis for a Subspace**
Consider the following subspace of \(\mathbb{R}^4\):
\[
\left\{
\begin{pmatrix}
9a + 18b - 3c \\
3a - b - c \\
-6a + 5b + 2c \\
-3a + b + c
\end{pmatrix}
: a, b, c \in \mathbb{R}
\right\}
\]
To determine a basis for this subspace, we are presented with the following options:
**Options:**
a. \(\{(9, 3, -6, 3), (18, -1, -5, 1)\}\)
b. \(\{(6, 3, -6, -3), (8, -1.5, -1)\}\)
c. \(\{(6, 3, -5, -3), (18, -1, 1.5, 1)\}\)
d. \(\{(9, 3, -6, 3), (9, -1, 5, 1)\}\)
e. \(\{(9, 3, -6, -3), (18, -1, 5, 1)\}\)
**Objective:**
Identify the correct pair of vectors that form a basis for the given subspace. To form a basis, the vectors must be linearly independent and span the subspace.
**Analysis:**
- The vector pairs provided represent sets of vectors in \(\mathbb{R}^4\).
- Evaluate linear independence by ensuring that no vector in the set can be written as a linear combination of the other vectors in the set.
- Ensure that the vectors span the subspace by ensuring that every vector in this subspace can be written as a linear combination of the vectors in the selected pair.
This problem invites students to apply their understanding of linear algebra concepts, particularly subspaces, bases, and linear independence, to find the appropriate set of vectors that define the basis for the given subspace.
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