Advanced Calculus fi..

Chapter 1 Vectors and Matrices 7Figure 1.6vectors u, v.(a) Linearly independent vectors u, v. (b) Linearly dependentFigure 1.7 (a) Linearly independent vectors u, v. w. (b) Linearly dependentvectors u, v, w.Figure 1.8 The vector uj as a linear combination of u,, u2,Conversely, if the three vectors are coplanar, then it can be verified that onemust equal a linear combination of the other two, say, u3 = klul + k2u2 and thenklul + kzu:! - lu3 = 0, SO the vectors are linearly dependent by the new definition.Again the two definitions agree.What about four vectors in space? Here the answer is simple: They must belinearly dependent. Let the vectors be ul, u:!, u3, u4. There are then two possibilities:(a) ul, u2, u3 are linearly dependent, and (b) u,, u2, us are linearly independent. Incase (a),for some scalars cl, c?. c not all 0. But thenwith not all of cl, ~ 2 , ~3 equal to 0. Thus ul. u2, ~ 3 . u4 are linearly dependent. Incase (b), ul, u2, U? are not coplanar and hence can be represented by the directededges of a parallelepiped in space, as in Fig. 1.9. From this it follows that u4 can