(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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(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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Recall:a limited number of vectors with linear dependence&amp;nbsp;{a1,a2,…,an&amp;nbsp;&amp;nbsp;pf a vector space V over a field F is said to be linearly dependent in V if there exist scalars&amp;nbsp;c1,c2,…cn, not all zero, in F such that&amp;nbsp;c1a1+c2a2+…+cnan=0&amp;nbsp;θ is the zero element of the vector space V.&amp;nbsp;(i)Given{v1,v2}is linearly independent i.e&amp;nbsp;v1,v2&amp;nbsp;are linearly dependent. Let 0 be the zero element. So there exist scalars&amp;nbsp;c1,c2&amp;nbsp;such that&amp;nbsp;c1v1+c2v2=0,&amp;nbsp;if any one of the scalar is 0,then the other scalar also becomess 0(since the vectors are non zero).&amp;nbsp;So for linear dependency both the scalars&amp;nbsp;c1,c2&amp;nbsp;have to be non-zero.Therefore,&amp;nbsp;c1v1+c2v2=0&amp;nbsp;⇒c1v1=−c2v2&amp;nbsp;⇒v1=(−c2c1)v2&amp;nbsp;⇒v1=cv2&amp;nbsp;Similarly,&amp;nbsp;c1v1+c2v2=0&amp;nbsp;⇒c2v2=−c1v1&amp;nbsp;⇒v2=(−c1c2)v1&amp;nbsp;⇒v2=cv1.&amp;nbsp;So it is proved that if&amp;nbsp;{v1,v2}&amp;nbsp;is linearly dependent, then there exist a constant c such that either&amp;nbsp;v1=cv2&amp;nbsp;or&amp;nbsp;v2=cv1.&amp;nbsp;(ii)Let there exist a constant c not equal to 0 such that either&amp;nbsp;v1=cv2(&amp;nbsp;or&amp;nbsp;v2=cv1).&amp;nbsp;Then it can be written as,&amp;nbsp;v1=cv2&amp;nbsp;⇒v1−cv2=0&amp;nbsp;Hence by the definition of linear dependence we can say that&amp;nbsp;v1,v2&amp;nbsp;are linearly dependent i.e.&amp;nbsp;{v1,v2}&amp;nbsp;is linearly dependent

(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

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