Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 4 - Vector Spaces - 4.4 Spanning Sets and Linear Independence - 4.4 Exercises - Page 178: 11


$S$ spans $R^2$.

Work Step by Step

Assume the combination $$a(5,0)+b(5,-4)=(0,0).$$ We have the system \begin{align*} 5a+5b&=0\\ -4b&=0. \end{align*} Since the determinant of the matrix is given by $$\left| \begin{array} {cc} 5&5\\0&-4 \end{array} \right|=-20\neq 0$$ then the system has unique solution, that is, $$a=0, \quad b=0.$$ Consequently, $S$ is linearly independent and since $R^2$ has the dimension $2$ then $S$ spans $R^2$.
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.