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Every vector in the set S is a linear combination of two vectors, (1, 1, 0) and (1, 0, 1), and every vector in the set K is a linear combination of two vectors, (1, 2, 1) and (1, 1, 2). Determine whether S K and explain your reason. (98 北科大車輛) Solution : Let u1 (1, 1, 0), u 2 (1, 0, 1), v1 (1, 2, 1), v 2 (1, 1, 2) Construct a matrix A with u1 , u 2 , v1 , v 2 as rows 1 0 1 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 R12 ( 1); R13 ( 1); R14 ( 1) R23 (1); R24 ( 2) 1 1 0 0 1 2 1 0 0 2 0 0 1 1 2 0 2 0 The rank of A is 2. v1 and v 2 are in the vector space of u1 and u 2 . Now consider av1 bv 2 0 ( a b, 2a b, a 2b) 0 a 0, b 0 This implies v1 and v 2 are linearly independen t . The space can be spanned by v1 and v 2 Hence S K . Linear System of Equations 1