Using row reduction, which of the vectors(keeping the same order) do we add to the set S=|so that we have a basis of R?Oa.O onlyb.O0 only1 onlyOd.10 onlyO e.O0 only1A NOO HO

Given vector space is R3(R) and dimension of R3(R) = 3.⇒ There are only three vectors which are in…

Answered: Using row reduction, which of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Using row reduction, which of the vectors
(keeping the same order) do we add to the set S=|
so that we have a basis of R?
Oa.
O only
b.O
0 only
1 only
Od.1
0 only
O e.O
0 only
1
A NO
O HO

Expert Solution

option (a) and option (b) are answer. For detail solution see the further.

$G i v e n v e c t o r s p a c e i s R^{3} (R) a n d d i m e n s i o n o f R^{3} (R) = 3 . \Rightarrow T h e r e a r e o n l y t h r e e v e c t o r s w h i c h a r e i n t h e b a s i s s e t . A n d a l l v e c t o r s i n b a s i s s e t a r e L i n e a r l y i n d e p e n d e n t . \Rightarrow o p t i o n (d) a n d (e) a r e n o t t h e a n s w e r b e c a u s e o p t i o n h a s o n l y o n e v e c t o r a n d i n c l u d i n g S t h e r e a r e t w o v e c t o r s . S o t h e y a r e n o t p o s s i b l e f o r b a i s i s o f v e c t o r s p a c e . o p t i o n (a) - I f w e c o n s i d e r [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 0 \end{matrix}] . T h e r e a r e t h r e e v e c t o r a n d a l l t h e s e a r e l i n e a r l y i n d e p e n d e n t, t h e n t h e y f o r m t h e b a s i s o f R^{3} . F o r t h i s w e c o n s i d e r a m a t r i x A, w h o s e r o w a r e t h e a b o v e v e c t o r s . A = [\begin{matrix} 1 & 0 & 1 \\ 0 & 0 & 2 \\ 0 & 1 & 0 \end{matrix}] \overset{R_{2} \leftrightarrow R_{3}}{\to} [\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{matrix}] \overset{R_{3} \to \frac{1}{2} R_{3}}{\to} [\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}]$

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