7. In the following, vi, V2, va) is a collection of vectors in a finite-dimensio

Question

7. In the following, vi, V2, va) is a collection of vectors in a finite-dimensional vector space V. Determine whether each of the following statements is true or false. If the statement is true, explain why. If the statement is false, come up with an example that shows it is false. (a)I i, 2, vs) spans V, then dim(V) s 3. (b) IE (vi. vs, vo) does not span V, then dim)2 (c) If V2, Va) is linearly dependent in V, then dim(V) s3. d)If ,v2, va) is linearly independent in V, then dim(V) 23. nust be a sp anning (f) If dim(V) = 3, then a set of more than three vectors must be linearly dependent.

Explanation / Answer

a)   If there exists a set {v1, V2.,v3} that spans V , then dim V 3

True. The Spanning Set Theorem says that we may remove elements from a spanning set to obtain a basis, so a spanning set must contain at least as many elements as the dimension of the space..

b) True because when vector does not span V ,have requried to at list one more element to get spanning set to obtain the basis.

c) False: Take V=R4 vetor so that V1=(1,0,0,0,V2=(0,1,0,0,V3=(1,1,0,0) are Linear dependent but not have basis .

d) Ture:

e)  True. A basis for V has p elements, and adding any element will not change the fact that the set spans.

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