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Q56E

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Linear Algebra With Applications

Found in: Page 177

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Show that the space of infinite sequence of real numbers is infinite dimensional.

The solution is the space V of infinite sequence is infinite dimensional.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Explanation of the solution

Assume that the opposite of that the space V of infinite sequences is finite dimensional with as follows.

dim(V)=n.

Consider an example of n+1 linearly independent infinite sequence as follows.

$(1, 0, 0, . . .), (0, 1, 00, . . .), (0, 0, 1, 0, . . .), . . ., (0, 0, 0, . . ., 0, 1, 0, . . .,)$ .

Step 2: Draw the conclusion

This contradicts the fact that n is the largest possible dimension for an n-dimensional space that m=n.

Therefore, the assumption is wrong.

Thus, the space V of infinite sequence is infinite dimensional.

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