chapter3.page15

chapter3.page15 - independent. • If function is one of f...

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3. LINEARLY INDEPENDENCY 15 Example 3.1 . For x 1 = 2 3 4 , x 2 = 0 1 - 4 , and x 3 = 4 8 0 , we can form a matrix, A = 2 3 4 0 1 - 4 4 8 0 , apply rref (type rref and in the place holder type A , then press =), rref ( A ) = 1 0 8 0 1 - 4 0 0 0 . We see that the vectors are linearly dependent as the last row is entirely zero. 3.2. Linearly independency of functions. We can also define linearly independency for a group of functions over an given inter- val [a, b]. Let f 1 ,f 2 , ··· ,f n be n functions defined over [a, b], C 1 ,C 2 , ··· ,C n are n scalars(numbers), the expression C 1 f 1 + C 2 f 2 + ··· + C n f n is called a linear combination of functions f 1 ,f 2 , ··· ,f n . Definition 3.2 . n functions f 1 ,f 2 , ··· ,f n is linearly indepen- dent over [a, b]if C 1 f 1 + C 2 f 2 + ··· + C n f n = 0 for all a t b (1) leads to C 1 = 0 ,C 2 = 0 , ··· ,C n = 0 . A set of function are linearly dependent if they are not linearly
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Unformatted text preview: independent. • If function is one of f 1 ,f 2 , ··· ,f n , then they linearly de-pendent. • Two nonzero functions f ( t ) and g ( t ) are linearly dependent over [a, b] if and only if f ( t ) = sg ( t ) for a constant s 6 = 0 and all a ≤ t ≤ b , for example, f ( t ) = t and g ( t ) = 4 t are linearly dependent but f ( t ) = t and g ( t ) = 4 t 2 are not, even f (0) = 4 g (0) and f (1) = 4 g (1) . • There are exists infinite many functions that are linearly in-dependent. For example the set { 1 ,t,t 2 ,t 3 , ··· ,t n , ···} is a linearly independent set....
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This note was uploaded on 10/04/2011 for the course MAP 4231 taught by Professor Dr.han during the Spring '11 term at UNF.

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