**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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Answered: 18. -5 | bartleby

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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