(i)Prove that if{v1,v2}is linearly dependent, then are multiple of each other, that is, there exists a constant c such that v1 = c v2 or v2=cv1. (ii)P

Trent Carpenter 2020-11-06 Answered

(i)Prove that ifv1,v2is linearly dependent, then are multiple of each other, that is, there exists a constant c such that v1=cv2 or v2=cv1.
(ii)Prove that the converse of(i) is also true.That is to say, if there exists a constant c such that v1=cv2 or v2=cv1, 1. thenv1,v2is linearly dependent.

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hosentak
Answered 2020-11-07 Author has 100 answers
Recall:Linear Dependence:A finite set of vectors {a1,a2,,an pf a vector space V over a field F is said to be linearly dependent in V if there exist scalars c1,c2,cn, not all zero, in F such that
c1a1+c2a2++cnan=0
θ is the zero element of the vector space V.
(i)Given{v1,v2}is linearly independent i.e v1,v2 are linearly dependent. Let 0 be the zero element. So there exist scalars c1,c2 such that
c1v1+c2v2=0,
if any one of the scalar is 0,then the other scalar also becomess 0(since the vectors are non zero).
So for linear dependency both the scalars c1,c2 have to be non-zero.Therefore,
c1v1+c2v2=0
c1v1=c2v2
v1=(c2c1)v2
v1=cv2
Similarly,
c1v1+c2v2=0
c2v2=c1v1
v2=(c1c2)v1
v2=cv1.
So it is proved that if {v1,v2} is linearly dependent, then there exist a constant c such that either v1=cv2orv2=cv1.
(ii)Let there exist a constant c not equal to 0 such that either v1=cv2(orv2=cv1).
Then it can be written as,
v1=cv2
v1cv2=0
Hence by the definition of linear dependence we can say that v1,v2 are linearly dependent i.e.
{v1,v2} is linearly dependent
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