Answered: Not a Vector Space Show lhatthe set Z…

Not a Vector Space Show lhatthe set Z of integers (stan- dard operations) is not a vector space by identifying at least one vector space property lhat fails.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...

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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

Not a Vector Space Show lhatthe set Z of integers (stan- dard operations) is not a vector space by identifying at least one vector space property lhat fails.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Determine if each of the following sets with the indicated operations of addition and scalar multiplication are a vector space. If so, verify the axioms, and if not, indicate at least one axiom that fails. i) Let V be the set of all positive real numbers. Define the operations of addition and scalar multiplication Ⓒ by x + y = xy and x©y=x² where the operations on the right side of each equation is usual real number multiplication and exponen- tial, respectively. ii) Let V denote the vectors in R2 with the usual vector addition, but scalar multiplication is instead defined as I -D-C с =
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an Given that { 91, 92, 93, 94} is a linearly independent set of vectors in av Vector space space V'( note that V need not be IR"). Use the diff Prove that {a an ax; -91 +92, 91-92 +93 ;-9, +92-93 +94"} is also a independence) linearly independent set (use the definition of Linearly
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