Problem 5. Let R[r] be the set of polynomials in x with real coefficients, and define the degree of a polynomial to be the highest power of x which has a nonzero coefficient. (Here x is a variable, not a vector.) Define the sets V = {p(x) € R[x]| degree(p(x)) < 3}, V2 = {p(x) E R[x]| degree(p(x)) = 3}, V3 = {p(x) E R[x] |p(0) = 0} For each of the three sets, show whether or not it (a) is closed under addition, (b) is closed under real scalar multiplication, (c) contains an additive identity (zero vector), (d) contains the additive inverse of each of its elements.

Problem 5. Let R[r] be the set of polynomials in x with real coefficients, and define the degree of a polynomial to be the highest power of x which has a nonzero coefficient. (Here x is a variable, not a vector.) Define the sets V = {p(x) € R[x]| degree(p(x)) < 3}, V2 = {p(x) E R[x]| degree(p(x)) = 3}, V3 = {p(x) E R[x] |p(0) = 0} For each of the three sets, show whether or not it (a) is closed under addition, (b) is closed under real scalar multiplication, (c) contains an additive identity (zero vector), (d) contains the additive inverse of each of its elements.

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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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.

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Problem 5. Let R[r] be the set of polynomials in x with real coefficients, and define the
degree of a polynomial to be the highest power of x which has a nonzero coefficient. (Here
x is a variable, not a vector.) Define the sets
V = {p(x) € R[x]| degree(p(x)) < 3},
V2 = {p(x) E R[x]| degree(p(x)) = 3},
V3 = {p(x) E R[x] |p(0) = 0}
For each of the three sets, show whether or not it
(a) is closed under addition,
(b) is closed under real scalar multiplication,
(c) contains an additive identity (zero vector),
(d) contains the additive inverse of each of its elements.

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