1. For a vector space V and a finite set of vectors S = {V₁,,Vn} in V, copy down the definitions for a) span(S) b) a basis for V c) a subspace of V
1. For a vector space V and a finite set of vectors S = {V₁,,Vn} in V, copy down the definitions for a) span(S) b) a basis for V c) a subspace 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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Help with question 1
![-
1. For a vector space V and a finite set of vectors S
definitions for
a) span(S)
(61
(1)
b) a basis for V
c) a subspace of V
(2) koga yba nielupes
2. Let V R³. Show that V with the given operations for and is not a vector
space. Clearly explain what goes wrong in terms of at least one of the axioms for
vector spaces.
stealbi
and
=
30
a) Verify that e₁
=
]
X1
is a basis for V = R³.
Yı
Z1
X1
8- -1
Yı
(c)x₁
(2c)y₁
(3c) z₁
21
(Hint: The addition is standard. Examine axiom 10 in the definition of a vector space)
3. Let
W:
{[₁] :* + P² <1}
be a subset of V=R2 with the standard addition and scalar multiplication.
[3],
[8]
b) Compute e₁ + e2, and show that it is NOT in W.
c) Explain why W is NOT then a subspace of V.
4. Explain why the set
yd bud
X2
Y/2
and e₂ =
Z2
{V₁, Vn} in V, copy down the
;
B =
x1 + x₂
Yı₁ + y2
21 +22]
are in W.
H
5
--(8-8-8)
0
0
(+) tacita
up Bolinti](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8783cbc5-0f10-43ba-8bd7-4625ee264f91%2Fd64988e3-c511-47e8-aecb-947cc6629552%2Fj04d5x_processed.jpeg&w=3840&q=75)
Transcribed Image Text:-
1. For a vector space V and a finite set of vectors S
definitions for
a) span(S)
(61
(1)
b) a basis for V
c) a subspace of V
(2) koga yba nielupes
2. Let V R³. Show that V with the given operations for and is not a vector
space. Clearly explain what goes wrong in terms of at least one of the axioms for
vector spaces.
stealbi
and
=
30
a) Verify that e₁
=
]
X1
is a basis for V = R³.
Yı
Z1
X1
8- -1
Yı
(c)x₁
(2c)y₁
(3c) z₁
21
(Hint: The addition is standard. Examine axiom 10 in the definition of a vector space)
3. Let
W:
{[₁] :* + P² <1}
be a subset of V=R2 with the standard addition and scalar multiplication.
[3],
[8]
b) Compute e₁ + e2, and show that it is NOT in W.
c) Explain why W is NOT then a subspace of V.
4. Explain why the set
yd bud
X2
Y/2
and e₂ =
Z2
{V₁, Vn} in V, copy down the
;
B =
x1 + x₂
Yı₁ + y2
21 +22]
are in W.
H
5
--(8-8-8)
0
0
(+) tacita
up Bolinti
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