Linear Algebra and Its Applications, 4th Edition

Published by Brooks Cole
ISBN 10: 0030105676
ISBN 13: 978-0-03010-567-8

Chapter 1 - Section 1.2 - The Geometry of Linear Equations - Problem Set - Page 9: 3

Answer

i) If $u\ne-1$ then we have a line in 4-dimensional space. ii) If $u= -1$ then we have a dot in 4-dimensional space.

Work Step by Step

Let, $u+v+w+z=6$ .............................$(1)$ $u+w+z=4$ .............................$(2)$ $u+w=2$ .............................$(3)$ Now, Subtract $(3)$ from $(2)$, $(2)$ from $(1)$ we get, $v+z=4$ .............................$(4)$ $z=2$ .............................$(5)$ $u+w=2$ .............................$(6)$ similarly from equation $(4),(5)$ we get, $v=2$ .............................$(7 )$ From above equations only equation $(6)$ is dependent. Hence, we can say that it is 4-dimensional space line. Now, put $u= -1$ in equation $(7 )$ we get, $u= -1$ $v=2$ $w=3$ $z=2$ and it is a dot in 4-dimensional space.
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