18. -5
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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
Transcribed Image Text:**Determine which sets in Exercises 15–20 are bases for \( \mathbb{R}^2 \) or \( \mathbb{R}^3 \). Justify each answer.**
In this task, you will analyze the sets provided in Exercises 15 through 20 to ascertain whether they can be considered bases for the vector spaces \( \mathbb{R}^2 \) or \( \mathbb{R}^3 \). Carefully evaluate each set and provide justification for your conclusions.
A base for a vector space must satisfy two criteria:
1. The set must be linearly independent.
2. The set must span the vector space.
For \( \mathbb{R}^2 \), you should have 2 vectors that meet these criteria.
For \( \mathbb{R}^3 \), you should have 3 vectors that fulfill these conditions.
![### Problem 18
Consider the following vectors given below:
\[
\begin{pmatrix}
1 \\
1 \\
-2 \\
\end{pmatrix}, \quad
\begin{pmatrix}
-5 \\
-1 \\
2 \\
\end{pmatrix}, \quad
\begin{pmatrix}
7 \\
0 \\
-5 \\
\end{pmatrix}
\]
These are three discrete 3-dimensional vectors. Let's examine each vector in detail:
1. The first vector \( \mathbf{v}_1 \) is
\[
\begin{pmatrix}
1 \\
1 \\
-2 \\
\end{pmatrix}
\]
2. The second vector \( \mathbf{v}_2 \) is
\[
\begin{pmatrix}
-5 \\
-1 \\
2 \\
\end{pmatrix}
\]
3. The third vector \( \mathbf{v}_3 \) is
\[
\begin{pmatrix}
7 \\
0 \\
-5 \\
\end{pmatrix}
\]
These vectors can be represented in a Cartesian coordinate system or used in operations involving vector arithmetic, spanning vector spaces, or in analyzing vector fields.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57103f69-f6d8-477a-b236-0336011ee35d%2F21ccb7c6-6b60-45b0-967d-5880ebd3b26d%2F6mkxzdb_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 18
Consider the following vectors given below:
\[
\begin{pmatrix}
1 \\
1 \\
-2 \\
\end{pmatrix}, \quad
\begin{pmatrix}
-5 \\
-1 \\
2 \\
\end{pmatrix}, \quad
\begin{pmatrix}
7 \\
0 \\
-5 \\
\end{pmatrix}
\]
These are three discrete 3-dimensional vectors. Let's examine each vector in detail:
1. The first vector \( \mathbf{v}_1 \) is
\[
\begin{pmatrix}
1 \\
1 \\
-2 \\
\end{pmatrix}
\]
2. The second vector \( \mathbf{v}_2 \) is
\[
\begin{pmatrix}
-5 \\
-1 \\
2 \\
\end{pmatrix}
\]
3. The third vector \( \mathbf{v}_3 \) is
\[
\begin{pmatrix}
7 \\
0 \\
-5 \\
\end{pmatrix}
\]
These vectors can be represented in a Cartesian coordinate system or used in operations involving vector arithmetic, spanning vector spaces, or in analyzing vector fields.
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