Prove that a nonempty subset U of a vector space Vover a field F is a subspace of V if, for every u and u' in U and everya in F, u + u' ∈ U and au ∈U. (In words, a nonempty set U isa subspace of V if it is closed under the two operations of V.)

Prove that a nonempty subset U of a vector space Vover a field F is a subspace of V if, for every u and u' in U and everya in F, u + u' ∈ U and au ∈U. (In words, a nonempty set U isa subspace of V if it is closed under the two operations of V.)

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

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

Prove that a nonempty subset U of a vector space V
over a field F is a subspace of V if, for every u and u' in U and every
a in F, u + u' ∈ U and au ∈U. (In words, a nonempty set U is
a subspace of V if it is closed under the two operations of V.)

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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