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 $(a, ar, {ar}^{2}, {ar}^{3}, . . . .)$ , for some constants and K.

The geometric sequences [i.e., sequences of the form $(a, ar, {ar}^{2}, {ar}^{3}, . . . .)$ , for some constants and K are subspaces of V .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 )

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, $X = \{(a, a r, a r^{2}, a r^{3}, . . .) | a, k \in R\} .$

Consider two arbitrary elements from X namely x and y.

Then, $x = (a, a r, a r^{2}, . . .), y = (b, b s, b s^{2}, . . .) .$

Find $x + y .$

$x + y = (a, a r, a r^{2}, . . .), (b, b s, b s^{2}, . . .) . = (a + b, a r + b s, a r^{2} + b s^{2}, \dots)$

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

$\frac{a r + b s}{a + b} = \frac{a r^{2} + b s^{2}}{a r + b s}$

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

Then, left side can be expressed as

$\frac{1.3 + 2.4}{1 + 2} = \frac{11}{2}$

Right side can be expressed as

$\frac{1 . 3^{2} + 2 . 4^{2}}{1.3 + 2.4} = \frac{9 + 32}{3 + 8} = \frac{41}{11}$

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

Therefore, it is not a subspace of V.

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Linear Algebra With Applications

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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 $(a, ar, {ar}^{2}, {ar}^{3}, . . . .)$ , for some constants and K.

Step by Step Solution

Step 1: Definition of subspace.

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

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Linear Algebra With Applications

Answers without the blur.

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.

Step by Step Solution

Step 1: Definition of subspace.

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

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Pure Maths

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 $(a, ar, {ar}^{2}, {ar}^{3}, . . . .)$ , for some constants and K.