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Transcript

i i i âmainâ 2007/2/16 page 271 i 4.5 Linear Dependence and Linear Independence 271 2. Any set of vectors in V containing the zero vector is linearly dependent. Remark We emphasize that the ï¬rst result in Proposition 4.5.7 holds only for the case of two vectors. It cannot be applied to sets containing more than two vectors. Example 4.5.8 If v1 = (1, 2, â9) and v2 = (â2, â4, 18), then {v1 , v2 } is linearly dependent in R3 , since v2 = â2v1 . Geometrically, v1 and v2 lie on the same line. Example 4.5.9 If A1 = 2 1 , 3 4 A2 = 0 0 , 0 0 A3 = 2 5 , â3 2 then {A1 , A2 , A3 } is linearly dependent in M2 (R), since it contains the zero vector from M2 (R). For more complicated situations, we must resort to Deï¬nitions 4.5.3 and 4.5.4, although conceptually it is always helpful to keep in mind that the essence of the problem we are solving is to determine whether a vector in a given set can be expressed as a linear combination of the remaining vectors in the set. We now give some examples to illustrate the use of Deï¬nitions 4.5.3 and 4.5.4. Example 4.5.10 If v1 = (1, 2, â1) v2 = (2, â1, 1), and v3 = (8, 1, 1), show that {v1 , v2 , v3 } is linearly dependent in R3 , and determine the linear dependency relationship. Solution: We must ï¬rst establish that there are values of the scalars c1 , c2 , c3 , not all zero, such that c1 v1 + c2 v2 + c3 v3 = 0. (4.5.1) Substituting for the given vectors yields c1 (1, 2, â1) + c2 (2, â1, 1) + c3 (8, 1, 1) = (0, 0, 0). That is, (c1 + 2c2 + 8c3 , 2c1 â c2 + c3 , âc1 + c2 + c3 ) = (0, 0, 0). Equating corresponding components on either side of this equation yields c1 + 2c2 + 8c3 = 0, 2c1 â c2 + c3 = 0, âc1 + c2 + c3 = 0. The reduced row-echelon form of the augmented matrix of this system is ï£® ï£¹ 1 0 2 0 ï£°0 1 3 0ï£». 0 0 0 0 Consequently, the system has an inï¬nite number of solutions for c1 , c2 , c3 , so the vectors are linearly dependent. In order to determine a speciï¬c linear dependency relationship, we proceed to ï¬nd c1 , c2 , and c3 . Setting c3 = t, we have c2 = â3t and c1 = â2t. Taking t = 1 and i i i i