1. Check that the following are vector spaces by proving in full detail that all axioms (one by one) in the definition of avector space are true.(a) The quadruple (R, R", H, D). Here, H and O denote the usual operations of addition and scalar multiplicationamong vectors.(b) The quadruple (R, R" [x], , 0), where R"[x] denotes the set of polynomials in one variable of degree less than orequal to n and with real coefficients. Here, H and O denote the usual operations of addition and scalar multipli-cation among polynomials.(c) The quadruple (R, C° (I), H, 0), where C° (I) denotes the set of real-valued continuous functions over an intervalI. Here, H and O denote the usual operations of addition and scalar multiplication among functions.

1. a) Let u, v, w ∈Rn ; so u= (u1,u2,u3,......,un) ,v=(v1,v2,v3,........,vn), w=…

Check that the following are vector spaces by proving in full detail that all axioms (one by one) in the definition of a vector space are true. (a) The quadruple (R, R",H, O). Here, and O denote the usual operations of addition and scalar multiplication among vectors.

Check that the following are vector spaces by proving in full detail that all axioms (one by one) in the definition of a vector space are true. (a) The quadruple (R, R",H, O). Here, and O denote the usual operations of addition and scalar multiplication among vectors.

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

1. Check that the following are vector spaces by proving in full detail that all axioms (one by one) in the definition of a
vector space are true.
(a) The quadruple (R, R", H, D). Here, H and O denote the usual operations of addition and scalar multiplication
among vectors.
(b) The quadruple (R, R" [x], , 0), where R"[x] denotes the set of polynomials in one variable of degree less than or
equal to n and with real coefficients. Here, H and O denote the usual operations of addition and scalar multipli-
cation among polynomials.
(c) The quadruple (R, C° (I), H, 0), where C° (I) denotes the set of real-valued continuous functions over an interval
I. Here, H and O denote the usual operations of addition and scalar multiplication among functions.

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