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i i i âmainâ 2007/2/16 page 270 i 270 CHAPTER 4 Vector Spaces Theorem 4.5.6 Let {v1 , v2 , . . . , vk } be a set of at least two vectors in a vector space V . Then {v1 , v2 , . . . , vk } is linearly dependent if and only if at least one of the vectors in the set can be expressed as a linear combination of the others. Proof If {v1 , v2 , . . . , vk } is linearly dependent, then according to Deï¬nition 4.5.3, there exist scalars c1 , c2 , . . . , ck , not all zero, such that c1 v1 + c2 v2 + Â· Â· Â· + ck vk = 0. Suppose that ci = 0. Then we can express vi as a linear combination of the other vectors as follows: vi = â 1 (c1 v1 + c2 v2 + Â· Â· Â· + ciâ1 viâ1 + ci+1 vi+1 + Â· Â· Â· + ck vk ). ci Conversely, suppose that one of the vectors, say, vj , can be expressed as a linear combination of the remaining vectors. That is, vj = c1 v1 + c2 v2 + Â· Â· Â· + cj â1 vj â1 + cj +1 vj +1 + Â· Â· Â· + ck vk . Adding (â1)vj to both sides of this equation yields c1 v1 + c2 v2 + Â· Â· Â· + cj â1 vj â1 â vj + cj +1 vj +1 + Â· Â· Â· + ck vk = 0. Since the coefï¬cient of vj is â1 = 0, the set of vectors {v1 , v2 , . . . , vk } is linearly dependent. As far as the minimal-spanning-set idea is concerned, Theorems 4.5.6 and 4.5.2 tell us that a linearly dependent spanning set for a (nontrivial) vector space V cannot be a minimal spanning set. On the other hand, we will see in the next section that a linearly v3 independent spanning set for V must be a minimal spanning set for V . For the remainder v2 of this section, however, we focus more on the mechanics of determining whether a given v1 set of vectors is linearly independent or linearly dependent. Sometimes this can be done x by inspection. For example, Figure 4.5.2 illustrates that any set of three vectors in R2 is linearly dependent. Figure 4.5.2: The set of vectors As another example, let V be the vector space of all functions deï¬ned on an interval {v1 , v2 , v3 } is linearly dependent I . If 2 in R , since v3 is a linear f1 (x) = 1, f2 (x) = 2 sin2 x, f3 (x) = â5 cos2 x, combination of v1 and v2 . y then {f1 , f2 , f3 } is linearly dependent in V , since the identity sin2 x + cos2 x = 1 implies that for all x â I , f1 (x) = 21 f2 (x) â 15 f3 (x). We can therefore conclude from Theorem 4.5.2 that span{1, 2 sin2 x, â5 cos2 x} = span{2 sin2 x, â5 cos2 x}. In relatively simple examples, the following general results can be applied. They are a direct consequence of the deï¬nition of linearly dependent vectors and are left for the exercises (Problem 49). Proposition 4.5.7 Let V be a vector space. 1. Any set of two vectors in V is linearly dependent if and only if the vectors are proportional. i i i i