Linear Algebra and Its Applications (5th Edition)

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

Chapter 1 - Linear Equations in Linear Algebra - 1.7 Exercises - Page 62: 23

Answer

$ \begin{bmatrix} \blacksquare&*&*\\ 0&\blacksquare&*\\ 0&0&\blacksquare \end{bmatrix} $

Work Step by Step

Recall that $\blacksquare$ represents a pivot and $*$ represents any numerical entry, including $0$. A matrix has linearly independent columns if and only if it contains a pivot in every column, leaving no free variables in the corresponding homogeneous matrix equation. By the rules defining an echelon matrix, the result must have the form above.
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