Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Show that in an n-dimensional linear space we can find at most n linearly independent elements.

The solution is m=n

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Statement of the theorem

If a linear space V has a basis with n elements, then all other bases of V consist of n elements as well.

We say that n is the dimension of V that is dim(V)=n.

Step2: Explanation of the solution

Consider two basis of V for a n-dimensional spaces as follows.

$B_{1} = (f_{1}, f_{2}, . . ., f_{n}) where \dim (B_{1}) = n B_{2} = (g_{1}, g_{2}, . . ., g_{m}) where \dim (B_{2}) = m$

To show that n is the largest possible dimension for an n-dimensional space that m=n.

Since, the m vectors of $B_{2}$ form a basis.

We know that as follows.

$c_{1} g_{1} + . . . + c_{m} g_{m} = 0$ , By the definition of linear independence.

Step 2: Draw the conclusion

Also know that $B_{1}$ forma a basis for the same space with n vectors and since the dimension of the space can’t change and as follows.

$m = n$

Since, m=n, that is there is no such basis m exists where $n \leq m$ for an n-dimensional space.

Thus, n is the largest possible dimension for an n-dimensional space that m=n

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Linear Algebra With Applications

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Short Answer

Show that in an n-dimensional linear space we can find at most n linearly independent elements.

Step by Step Solution

Step1: Statement of the theorem

Step2: Explanation of the solution

Step 2: Draw the conclusion

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Linear Algebra With Applications

Answers without the blur.

Short Answer

Show that in an n-dimensional linear space we can find at most n linearly independent elements.

Step by Step Solution

Step1: Statement of the theorem

Step2: Explanation of the solution

Step 2: Draw the conclusion

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