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Q15E

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Linear Algebra and its Applications
Found in: Page 191
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

If A is a \({\bf{6}} \times {\bf{8}}\) matrix, what is the smallest possible dimension of Null A?

The smallest possible dimension of Null A is 2.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Describe the given data

From the given \(6 \times 8\) matrix, the number of pivots cannot exceed 6. That is

\({\rm{rank}}\,A \le 6\).

Step 2: Use the rank theorem

By the rank theorem, you get

\(\begin{aligned} n &= {\rm{rank}}\,A + \dim \,{\rm{Null}}\,A\\8 &\le 6 + \dim \,{\rm{Null}}\,A\\8 - 6 &\le \dim \,{\rm{Null}}\,A\\2 &\le \dim \,{\rm{Null}}\,A.\end{aligned}\)

Step 3: Draw a conclusion

Hence, the smallest possible dimension of Null A is 2.

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