4. The set S = {ão + a₁x + a₂x²: ao + a₁ + a₂ = 0}.
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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Question
can you please work on number 4 and explain step by step! SHOW YOUR WORK ON PICTURES, PLEASe!
Transcribed Image Text:and scalar multiplication, forms a vector space.
(i) The set of positive real numbers, with the usual op-
erations of addition and scalar multiplication, forms a
vector space.
(j) The set of nonnegative real numbers (i.e., {x € R :
x ≥ 0}), with the usual operations of addition and
scalar multiplication, forms a vector space.
Problems
For Problems 1-14, determine whether the given set S of
vectors is closed under addition and closed under scalar mul-
tiplication. In each case, take the set of scalars to be the set
of all real numbers.
1. The set S := Q of all rational numbers. ¹
2. The set S = Un(R) of all upper triangular n × n
matrices with real elements.
3. The set
S := {A € M₂ (R): A is upper or lower triangular}.
4. The set S := {ao + a₁x + a₂x² : ao + a₁ + a2 = 0}.
5. The set S := {ao + a₁x + a2x² : ao + a₁ + a₂ = 1}.
6. The set S of all solutions to the differential equation
y' + 3y 6x³ + 5. (Do not solve the differential
equation.)
=
7. The set S of all solutions to the differential equation
y' + 3y = 0. (Do not solve the differential equation.)
14. The set S of all polynomials of the forma + bx³ + cx4,
where a, b, c ER.
=
15. We have defined the set R² {(x, y) : x, y ≤ R},
together with the addition and scalar multiplication
operations as follows:
(x1, y₁) + (x2, Y2) = (x1 + x2, y1 + y2),
k(x₁, y₁) = (kxı, ky₁).
Give a complete verification that each of the vector
space axioms is satisfied.
=
16. Determine the zero vector in the vector space V
M4x2 (R), and write down a general element A in V
along with its additive inverse - A.
17. Generalize the previous exercise to find the zero vec-
tor and the additive inverse of a general element of
Mmxn (R).
=
18. Determine the zero vector in the vector space V
P3 (R), and write down a general element p(x) in V
along with its additive inverse -p(x).
19. Generalize the previous exercise to find the zero vec-
tor and the additive inverse of a general element of
Pn (R).
20. On R+, the set of positive real numbers, define the op-
erations of addition, , and scalar multiplication, O,
as follows:
x + y =
= xy
COX = = xc.
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