Linear Algebra: A Geometric Approach - 2 Edition - Chapter 3.6 - Problem 6
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# Decide whether each of the following is a subspace of C0(R). If so, provide a basis and

Linear Algebra: A Geometric Approach | 2nd Edition

Problem 6

Decide whether each of the following is a subspace of C0(R). If so, provide a basis and determine its dimension. If not, give a reason. a. {f : f (1) = 2} b. {f P2 : + 1 0 f (t)dt = 0} c. {f C1(R) : f _ (t) + 2f (t) = 0 for all t} d. {f P4 : f (t) tf _ (t) = 0 for all t} e. {f P4 : f (t) tf _ (t) = 1 for all t} f. {f C2(R) : f __ (t) + f (t) = 0 for all t} g. {f C2(R) : f __ (t) f _ (t) 6f (t) = 0 for all t} h. {f C1(R) : f (t) = + t 0 f (s)ds for all t}

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Step-by-Step Solution:

Problem 6

Decide whether each of the following is a subspace of . If so, provide a basis and determine its dimension. If not, give a reason

a.

b.

c.

d.

e.

f.

g.

h.

Step by Step Solution

Step 1 of 8

(a)

Let us denote the given set by  i.e.

It is known that a non-empty subset  of a vector space  is a subspace of  if:

*

*

*

From the given definition of , it is clear that the zero function  doesn’t belong to  as it doesn’t satisfy the . Therefore,  is not a subspace.

###### Chapter 3.6, Problem 6 is Solved

Step 2 of 8

Step 3 of 8

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