**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 18Consider 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.

Name: **Determine which sets in Exercises 15–20 are bases for $ \mathbb{R}
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Description: **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

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Math

Advanced Math

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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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

Topic Video

Question

**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.

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