Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - 1.7 Exercises - Page 62: 28

Answer

The $5\times7$ matrix must have five pivot columns.

Work Step by Step

Suppose there were only four pivot columns. Then the matrix equation $\mathbf{A}\vec{x}=\vec{0}$ would have four basic variables and three free variables, i.e., it would have four linearly independent columns. But four linearly independent vectors span only a 4-dimensional subspace of $\mathbb{R}^{5}$. Hence, the matrix must have more than four pivot columns. But a matrix cannot have more pivot positions than there are rows, so the matrix must have exactly five pivot columns.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.