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2 MA 36600 LECTURE NOTES: FRIDAY, APRIL 10 Systems of Linear Equations Linear Independence. Consider a collection of m -dimensional vectors: x ( k ) = a 1 k a 2 k . . . a mk . We say that the set ° x (1) , x (2) ,. . . , x ( n ) ± is a linearly independent set if the only solution c k to the equation c 1 x (1) + c 2 x (2) + ··· + c n x ( n ) = 0 is c 1 = c 2 = ··· = c n = 0. Otherwise, we say that the set if linearly dependent . Given an m × n matrix A ,let x ( k ) denote the k th column as a vector. Then the following are equivalent: i. ° x (1) , x (2) ,. . . , x ( n ) ± is linearly independent. ii. A is nonsingular. Recall that A is nonsingular if ker( A )= { 0 } . In particular, if m = n ,thentheset ° ..., x ( k ) ,... ± is linearly independent if and only if det A ° = 0. We present proof of this equivalence. Denote the matrices A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a mn , x =
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