In Problems 19–24, use the method of Example 3 to determine whether the given vectors u, v, and w are linearly independent or dependent. If they are linearly dependent, find scalars a, b, and c not all zero such that au + bv + cw = 0.

u = (1 , 4 , 5) , v = (4 , 2 , 5) , w = (-3 , 3 , -1)

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Example 3 To determine whether the three vectors u = (1, 2,-3), v = (3, 1, -2), and w = (5,-5, 6) are linearly independent or dependent, we need to solve the system a []-[8] b By Gaussian elimination, we readily reduce this system to the echelon form 1 3 5 0 1 3 = -[8] 000 au + bv + cw = 1 2 -3-2 3 5 1 -5 6 a b C Therefore, we can choose c = 1, and it follows that b = -3c = −3 and a = -3b-5c = 4. Therefore, 4u + (-3)v + w = 0, and hence u, v, and w are linearly dependent, with w = -4u + 3v.