7. Let V = Ca, b] be the set of continuous functions over [a, b], where a, b. Let a
7. Let V = Ca, b] be the set of continuous functions over [a, b], where a, b. Let a
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
See picture, question 7
![given operations.
X +Ỷ = XỶ
cX = X°
if it is verify all 10 axioms; if not state all vector space axioms that fail.
3. Let V be the set of all fifth-degree polynomials with standard operations. Is it a vector space.
Justify your answer.
4. Let V = {(x,x)|x E R} with standard operations. Is it a vector space. Justify your answer.
5. Let V = R2. Let W = {(x, y)
y = mx}. Is VW a subspace? How about the line y = mx +6
6. Let V be the vector space of all 3X3 matrices. Let W = {A E VAexists}. Is w a vector
space.
7. Let V = Cla, b be the set of continuous functions over [a, b], where a, b. Let a <d <b and
let D = {f € V\f'(d)exists}. prove that V is a subspace.
8. Determine which of the following are subspace of M22;
(a) The set of matrices of the form
where a, b, c and d are integers;
(b) The ser of 2x2 matrices such that A = A";
(c) The set of 2x2 matrices such that det(A)= 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabc5aa9d-951a-40e1-86a3-f092042c4046%2Fae4e8842-397d-4e78-8378-090e063f0502%2Fscev1ja_processed.jpeg&w=3840&q=75)
Transcribed Image Text:given operations.
X +Ỷ = XỶ
cX = X°
if it is verify all 10 axioms; if not state all vector space axioms that fail.
3. Let V be the set of all fifth-degree polynomials with standard operations. Is it a vector space.
Justify your answer.
4. Let V = {(x,x)|x E R} with standard operations. Is it a vector space. Justify your answer.
5. Let V = R2. Let W = {(x, y)
y = mx}. Is VW a subspace? How about the line y = mx +6
6. Let V be the vector space of all 3X3 matrices. Let W = {A E VAexists}. Is w a vector
space.
7. Let V = Cla, b be the set of continuous functions over [a, b], where a, b. Let a <d <b and
let D = {f € V\f'(d)exists}. prove that V is a subspace.
8. Determine which of the following are subspace of M22;
(a) The set of matrices of the form
where a, b, c and d are integers;
(b) The ser of 2x2 matrices such that A = A";
(c) The set of 2x2 matrices such that det(A)= 0.
Expert Solution
Step 1
Given be the set of continuous functions on [a,b] and .
To show that is a subspace. (I think it is the question since )
Definition:
Let V be a vector space over a field F. A non-void subset S of V is a subspace of V iff for all u,v and for all
i. and .
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