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math-2011-note7 - Theorem 34(Thm 13.3 p.291 The general...

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Theorem 34 (Thm 13.3, p.291) . The general quadratic form Q ( x 1 , ··· , x k ) = i j a i j x i x j can be written as ( x 1 x 2 ··· x k ) a 11 1 2 a 12 ··· 1 2 a 1 k 1 2 a 12 a 22 ··· 1 2 a 2 k . . . . . . . . . . . . 1 2 a 1 k 1 2 a 2 k ··· a kk x 1 x 2 . . . x k , or x T A x , where A is a symmetric matrix. 11.2 Continuous Functions Defnition 24 (p.293) . A function f : R k R m is continuous at x 0 if whenever { x n } n = 1 converges to x 0 the sequence { f ( x n ) } n = 1 in R m converges to f ( x 0 ) . The function f is said to be continuous if it is continuous at every point in its domain. Theorem 35 (Thm 13.5, p.295) . Let f = ( f 1 , ··· , f m ) be a function from R k to R m . Then, f is continuous at x if and only if each of its component functions f i : R k R 1 is continuous at x . If f and g are two functions from R k to R m , f + g , f - g , f · g are deFned by ( f + g )( x ) = f ( x )+ g ( x ) , ( f - g )( x ) = f ( x ) - g ( x ) , ( f · g )( x ) = f ( x ) · g ( x ) , respectively. Theorem 36 (Thm 13.4, p.294) . Let f and g be functions from R k to R m . If f and g are continuous at x , then f
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