Measurements

Physics of the earth's space environment by Prolss

By Prolss

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B. Let S1 and S2 be finite subsets of vectors in Rn such that S1  S2 . If S1 is linearly independent, show by examples in R3 that S2 may be either linearly dependent or linearly independent. 11. Let S ¼ fv1 , v2 , . . , vk g be a set of vectors in Rn. Show that if one of these vectors is the zero vector, then S is linearly dependent. 12. a. Prove that any subset of a linearly independent set of vectors is linearly independent. b. Prove that any set of vectors containing a linearly dependent subset is linearly dependent.

Say that aiþ1 6¼ 0 (if not, just rearrange or relabel the terms in the second set of parentheses). Then &    ' wiþ1 a1 ai w1 þ Á Á Á þ wi À viþ1 ¼ aiþ1 aiþ1 aiþ1 &    ' aiþ2 am viþ2 þ Á Á Á þ vm , À aiþ1 aiþ1 which implies that fw1 , w2 , . . , wiþ1 , viþ2 , . . , vm g spans Rn. Thus we have proved the claim by induction. If k > m, eventually the vi ’s disappear, having been replaced by the wj ’s. Furthermore, span {w1, w2, . . , wm} ¼ Rn. Then each of the vectors wmþ1 , wmþ2 , . . , wk is a linear combination of w1 , w2 , .

In each space Rn, there are special sets of vectors that play an important ! 1 and role in describing the space. For example, in R2 the vectors e1 ¼ 0 ! 0 x have the significant property that any vector v ¼ can be e2 ¼ 1 y written ! as a linear combination of vectors e1 and e2 : x v¼ ¼ xe1 þ ye2 . In physics and engineering, the symbols i and j y (or~i and ~ j) are often used for e1 and e2 , respectively. 8). 8 The vectors e1 and e2 in R2. 2 3 x w ¼ 4 y 5 ¼ xe1 þ ye2 þ ze3 . In applied courses, these three vectors are z often denoted by i, j, and k (or by2~i, ~ j,3 and ~ k), respectively.

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