Number 2 part a and b and number 4 from 4.1 vector space 196CHAPTER 4Vector Spacesb. Find specific vectors u and v in W such that u + v isnot in W. This is enough to show that W is not a vectorIn Exercises 15-18, let W be the set of allshown, where a, b, and c represent arbitraryeach case, either find a set S of vectors that spaexample to show that W is not a vector space.space.3. Let H be the set of points inside and on the unit circle inthe xy-plane. That is, let H =:x + y < 1.2a + 3bFind15.-116.3a -5ba specific example-two vectors or a vector and a scalar-toshow that H is not a subspace of R².2a - 5b36+2a2a-b3b - c4a+3E4. Construct a geometric figure that illustrates why a line in R?not through the origin is not closed under vector addition.17.013c - a18.a+36 +36-2с19. If a mass m is placed at the end of a spring, anpulled downward and released, the mass-spribegin to oscillate. The displacement y of theresting position is given by a function of the faIn Exercises 5-8, determine if the given set is a subspace of P, foran appropriate value of n. Justify your answers.3b5. All polynomials of the form p(t) = at?, where a is in R.6. All polynomials of the form p(t) = a + 1², where a is in R.7. All polynomials of degree at most 3, with integers as coeffi-y(t) = c1 cos wt + c2 sin wtcients.where w is a constant that depends on the spring(See the figure below.) Show that the set ofdescribed in (5) (with w fixed and c,Cz arbitran8. All polynomials in P, such that p(0) = 0.-2tspace.9. Let H be the set of all vectors of the form5tFind a3tvector v in R3 such that H = Span {v}. Why does this showthat H is a subspace of R3?3t10. Let H be the set of all vectors of the formwhere t-7tis any real number. Show that H is a subspace of R. (Usethe method of Exercise 9.)[2b + 3c11. Let W be the set of all vectors of the form--b2c20. The set of all continuous real-valued functionsclosed interval [a,b] in R is denoted by Cla, b!a subspace of the vector space of all real-valuedefined on [a, b].a. What facts about continuous functions shouin order to demonstrate that C[a, b] is indeeas claimed? (These facts are usually discalculus class.)where b and c are arbitrary. Find vectors u and v such thatW = Span {u, v}. Why does this show that W is a subspaceof R3?2s+4t2.s12. Let W be the set of all vectors of the form2s - 315tShow that W is a subspace of R. (Use the method ofExercise 11.)b. Show that {f in Cla, b]: f(a) = f(b)} is aCla, b).4For fixed positive integers m and n, the set Mxematrices is a vector space, under the usual operationof matrices and multiplication by real scalars.13. Let vị =0 , V2 =2and w =1V3 =36.a. Is w in {v, V2, V3}? How many vectors are in {v1, V2, V3}?wwww Is, and they will receive more attention later.ASof all points in R? of the form (3s,2+5s) is not a vector space,not closed under scalar multiplication. (Find a specific vectorsuch that cu is not in H.)..,vp}, where V1,... , Vp are in a vector space V. Show that Vkp. [Hint: First write an equation that shows that vi is in W.cation for the general case.]that cu is not in V. (This is enough to show that V is nota vector space.)2. Let W be the union of the first and third quadrants in thexy-plane. That is, let W =: ху >uchIf u is in W and c is any scalar, is cu in W? Why?a.

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2. Let W be the union of the first and third quadrants in the xy- plane. That is, let W = xy 0 a. If u is in W and c is any scalar, is cu in W? Why?

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

Number 2 part a and b and number 4 from 4.1 vector space

196
CHAPTER 4
Vector Spaces
b. Find specific vectors u and v in W such that u + v is
not in W. This is enough to show that W is not a vector
In Exercises 15-18, let W be the set of all
shown, where a, b, and c represent arbitrary
each case, either find a set S of vectors that spa
example to show that W is not a vector space.
space.
3. Let H be the set of points inside and on the unit circle in
the xy-plane. That is, let H =
:x + y < 1.
2a + 3b
Find
15.
-1
16.
3a -5b
a specific example-two vectors or a vector and a scalar-to
show that H is not a subspace of R².
2a - 5b
36+2a
2a-b
3b - c
4a+3E
4. Construct a geometric figure that illustrates why a line in R?
not through the origin is not closed under vector addition.
17.
01
3c - a
18.
a+36 +
36-2с
19. If a mass m is placed at the end of a spring, an
pulled downward and released, the mass-spri
begin to oscillate. The displacement y of the
resting position is given by a function of the fa
In Exercises 5-8, determine if the given set is a subspace of P, for
an appropriate value of n. Justify your answers.
3b
5. All polynomials of the form p(t) = at?, where a is in R.
6. All polynomials of the form p(t) = a + 1², where a is in R.
7. All polynomials of degree at most 3, with integers as coeffi-
y(t) = c1 cos wt + c2 sin wt
cients.
where w is a constant that depends on the spring
(See the figure below.) Show that the set of
described in (5) (with w fixed and c,Cz arbitran
8. All polynomials in P, such that p(0) = 0.
-2t
space.
9. Let H be the set of all vectors of the form
5t
Find a
3t
vector v in R3 such that H = Span {v}. Why does this show
that H is a subspace of R3?
3t
10. Let H be the set of all vectors of the form
where t
-7t
is any real number. Show that H is a subspace of R. (Use
the method of Exercise 9.)
[2b + 3c
11. Let W be the set of all vectors of the form
--b
2c
20. The set of all continuous real-valued functions
closed interval [a,b] in R is denoted by Cla, b!
a subspace of the vector space of all real-value
defined on [a, b].
a. What facts about continuous functions shou
in order to demonstrate that C[a, b] is indee
as claimed? (These facts are usually dis
calculus class.)
where b and c are arbitrary. Find vectors u and v such that
W = Span {u, v}. Why does this show that W is a subspace
of R3?
2s+4t
2.s
12. Let W be the set of all vectors of the form
2s - 31
5t
Show that W is a subspace of R. (Use the method of
Exercise 11.)
b. Show that {f in Cla, b]: f(a) = f(b)} is a
Cla, b).
4
For fixed positive integers m and n, the set Mxe
matrices is a vector space, under the usual operation
of matrices and multiplication by real scalars.
13. Let vị =
0 , V2 =
2
and w =
1
V3 =
3
6.
a. Is w in {v, V2, V3}? How many vectors are in {v1, V2, V3}?
wwww

Is, and they will receive more attention later.
AS
of all points in R? of the form (3s,2+5s) is not a vector space,
not closed under scalar multiplication. (Find a specific vector
such that cu is not in H.)
..,vp}, where V1,... , Vp are in a vector space V. Show that Vk
p. [Hint: First write an equation that shows that vi is in W.
cation for the general case.]
that cu is not in V. (This is enough to show that V is not
a vector space.)
2. Let W be the union of the first and third quadrants in the
xy-
plane. That is, let W =
: ху >
uch
If u is in W and c is any scalar, is cu in W? Why?
a.

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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