Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 4 - Vector Spaces - 4.4 Spanning Sets - Problems - Page 282: 5

Answer

Yes.

Work Step by Step

We know that for a matrix \[ \left[\begin{array}{rrr} a & b & c \\ d &e & f \\ g &h & i \\ \end{array} \right] \] the determinant, $D=a(ei-fh)-b(di-fg)+c(dh-eg).$ Hence here $D=1(5-(-6))-(-1)(2-12)+1(-4-20)=1(11)+1(-10)+1(-24)=11-10-24=-23\ne0$ Thus they are not coplanar, they span the space.
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