Answered: Give an example of a subset (not…

Give an example of a subset (not subspace) of R3 that has an infinite number of elements, is closed under addition, contains the zero vector, but is not closed under scalar multiplication.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.3: Subspaces Of Vector Spaces

Problem 56E: Give an example showing that the union of two subspaces of a vector space V is not necessarily a...

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Give an example of a subset (not subspace) of R that has an infinite number of
elements, is closed under addition, contains the zero vector, but is not closed under scalar
multiplication.

Expert Solution

Step 1

A subset $W$ of vector space $V$ over the scalar field $K$ is a subspace of $V$ if and only if following three criteria are met.

$W$ contains the zero vector of $V$ .
If $u, v \in W$ , then $u + v \in W$ . then $W$ is closed under addition.
If $u \in W, a \in K$ then $a u \in W$ then $W$ is closed under scalar multiplication.

Given vector space is $R^{3}$ , here the scalar field is $R$ .

Let us define $W = {(x_{1}, x_{2}, x_{3}) \in R^{3} / x_{1} \geq 0, x_{2}, x_{3} \in R} .$

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