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Linear Algebra With Applications
Found in: Page 176
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Let V be the space of all infinite sequences of real numbers. See Example 5. Which of the subsets of given in Exercises 12 through 15 are subspaces of V? The geometric sequences [i.e., sequences of the form (a,ar,ar2,ar3,....), for some constants and K.

The geometric sequences [i.e., sequences of the form (a,ar,ar2,ar3,....), for some constants and K are subspaces of V .

See the step by step solution

Step by Step Solution

Step 1: Definition of subspace.

A subset W of a linear space V is called a subspace of V if

(a) W contains the neutral element 0 of V .

(b) W is closed under addition (if f and g are in W then so is f+g )

(c) W is closed under scalar multiplication (if f is in W and K is scalar, then kf is in W ).

we can summarize parts b and c by saying that W is closed under linear combinations.

Step 2: Verification whether the subset is closed under addition and scalar multiplication.

Let us consider a set X to be the set of all geometric sequences which is a subset of V.

Then, X=(a,ar,ar2,ar3,...)|a,kR.

Consider two arbitrary elements from X namely x and y.


Find x+y.

x+y=(a,ar,ar2,...),(b,bs,bs2,...). =(a+b,ar+bs,ar2+bs2,)

In a geometric sequence, on dividing two consecutive elements we get a common ratio. (i.e.)


Let us assume a=1,r=3, b=2and s=4.

Then, left side can be expressed as


Right side can be expressed as

1.32+2.421.3+2.4=9+323+8 =4111

It is obtained that L.H.S R.H.S. Thus, it is not closed under addition.

Therefore, it is not a subspace of V.

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