See picture, question 7 given operations.X +Ỷ = XỶcX = 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 +66. Let V be the vector space of all 3X3 matrices. Let W = {A E VAexists}. Is w a vectorspace.7. Let V = Cla, b be the set of continuous functions over [a, b], where a, b. Let a

Answered: 7. Let V = Ca, b] be the set of…

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7. Let V = Ca, b] be the set of continuous functions over [a, b], where a, b. Let a

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.6: Rank Of A Matrix And Systems Of Linear Equations

Problem 67E: Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many...

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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.

Expert Solution

Step 1

Given $C [a, b]$ be the set of continuous functions on [a,b] and $a < d < b$ .

To show that $D = \{f \in V / f' (d) e x i s t s\}$ is a subspace. (I think it is the question since $D \subset V$ )

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 $\in S$ and for all $a \in F$

i. $u - v \in S$ and $a u \in S$ .

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