TRANSCRIBE THE FOLLOWING TEXT IN DIGITAL FORMAT Sol:-Given thatNoweS = {(-3, 2)}(d) s= {(1,3,0), (4,1,2), (-2,5,-2)}2©сSO5= {(-3,2)} is not a basis of 1R²because, we knowdim (IR²) = 2,Now if S is albo basis of IR2then dim (IR²) = Number of element ins= 1which isa contradiction.S is not abasis of IR2d)S ={(1,3,0), (4,1,2), (-2,5,-2)}I43 1let A =-252 -2det (A) = 1x (-2-10) -4 (-6) -2 (6)= -12-24-12= - 48 0Hence s is a linearly dep independent setcontaing 3 vectors of IR³we know a famous result if A GIR" be alinearly independent set containg n vectersof Irn thenRnthen A is basis of RhByTherefore(Ⓒ)this results is abasis of IR³S is a basiss={(-3,2)} is not a basis of 1R2(d) S =S= {(1, 3,0), (4,1,2), (-2, 5, -2)} is abasis of IR³

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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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TRANSCRIBE THE FOLLOWING TEXT IN DIGITAL FORMAT

Sol:-
Given that
Now
e
S = {(-3, 2)}
(d) s= {(1,3,0), (4,1,2), (-2,5,-2)}
2
©
с
SO
5= {(-3,2)} is not a basis of 1R²
because, we know
dim (IR²) = 2,
Now if S is albo basis of IR2
then dim (IR²) = Number of element ins
= 1
which is
a contradiction.
S is not a
basis of IR2
d)
S =
{(1,3,0), (4,1,2), (-2,5,-2)}
I
4
3 1
let A =
-2
5
2 -2
det (A) = 1x (-2-10) -4 (-6) -2 (6)
= -12-24-12
= - 48 0
Hence s is a linearly dep independent set
containg 3 vectors of IR³
we know a famous result if A GIR" be a
linearly independent set containg n vecters
of Irn then
Rn
then A is basis of Rh
By
Therefore
(Ⓒ)
this results is a
basis of IR³
S is a basis
s={(-3,2)} is not a basis of 1R2
(d) S =
S
= {(1, 3,0), (4,1,2), (-2, 5, -2)} is a
basis of IR³

Expert Solution

Step 1: Introduction of the given problem

Given that

$open parentheses c close parentheses space S equals open curly brackets open parentheses negative 3 comma 2 close parentheses close curly brackets open parentheses d close parentheses space S equals open curly brackets open parentheses 1 comma 3 comma 0 close parentheses comma open parentheses 4 comma 1 comma 2 close parentheses comma open parentheses negative 2 comma 5 comma negative 2 close parentheses close curly brackets$