Exercise 2.A.11 Proof. - Stanford University

math 113 homework 2 SolutionsSolutions by Guanyang Wang, with edits by Tom from the , ..,vmis linearly independent inVandw thatv1, .., vm, wis linearly independent if and only ifw / span(v1, .., vm) supposev1, .., vm, wis linearly independent. Then ifw span(v1, .., vm),we can writewas the linear combination ofv1, .., vm, that isw=a1v1+..+ both sides of the equation by w, we havea1v1+..+amvm+ ( w) = 0 Therefore we can write 0 asa1v1+..+amvm+( w), so there existsa1, a2, .., am, 1,not all 0, such thata1v1+..+amvm+ ( w) = 0. by the definition of linear de-pendence, we havev1, ..vm, wis linearly dependent, which contradicts our initialassumption. Thus we havew / span(v1, .., vm).Conversely, supposew / span(v1, .., vm). Ifv1, .., vm, wis linearly dependent,then by the linear dependence lemma(Lemma ), we havevj span(v1, .., vj 1)for somejorw span(v1, .., vm). But sincev1, .., vmis linearly independent,there is noj {1.}

Math 113 Homework 2 Solutions Solutions by Guanyang Wang, with edits by Tom Church. Exercises from the book. Exercise 2.A.11 Suppose v 1, ..., v m is linearly independent in V and w 2V. Show that v 1;:::;v m;w is linearly independent if and only if w =2span(v

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