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lineqs - EE103 (Fall 2009-10) 3. Linear equations •...

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Unformatted text preview: EE103 (Fall 2009-10) 3. Linear equations • linear equations • example: polynomial interpolation • applications • geometrical interpretation • left and right inverse • range and nullspace 3-1 Linear equations m equations in n variables x 1 , x 2 , . . . , x n : a 11 x 1 + a 12 x 2 + ··· + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ··· + a 2 n x n = b 2 . . . a m 1 x 1 + a m 2 x 2 + ··· + a mn x n = b m in matrix form: Ax = b , where A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a mn , x = x 1 x 2 . . . x n , b = b 1 b 2 . . . b m Linear equations 3-2 Example: polynomial interpolation fit a polynomial p ( t ) = x 1 + x 2 t + x 3 t 2 + ··· + x n t n − 1 through n points ( t 1 ,y 1 ) , . . . , ( t n ,y n ) t 1 t 2 t 3 t 4 t 5 problem data: t 1 , . . . , t n , y 1 , . . . , y n ; problem variables: x 1 , . . . , x n Linear equations 3-3 write out the conditions on x : p ( t 1 ) = x 1 + x 2 t 1 + x 3 t 2 1 + ··· + x n t n − 1 1 = y 1 p ( t 2 ) = x 1 + x 2 t 2 + x 3 t 2 2 + ··· + x n t n − 1 2 = y 2 . . . p ( t n ) = x 1 + x 2 t n + x 3 t 2 n + ··· + x n t n − 1 n = y n in matrix form: Ax = b with A = 1 t 1 t 2 1 ··· t n − 1 1 1 t 2 t 2 2 ··· t n − 1 2 . . . . . . . . . . . . . . . 1 t n t 2 n ··· t n − 1 n , b = y 1 y 2 . . . y n n linear equations in n variables Linear equations 3-4 Exercise express as a set of linear equations: find two cubic polynomials p ( t ) = c + c 1 t + c 2 t 2 + c 3 t 3 , q ( t ) = d + d 1 t + d 2 t 2 + d 3 t 3 that satisfy the following properties: • p ( t 1 ) = y 1 , p ( t 2 ) = y 2 , p ( t 3 ) = y 3 ( t i , y i given for i = 1 , 2 , 3 ) • q ( t 5 ) = y 5 , q ( t 6 ) = y 6 , q ( t 7 ) = y 7 ( t i , y i given for i = 5 , 6 , 7 ) • p ( t 4 ) = q ( t 4 ) , p ′ ( t 4 ) = q ′ ( t 4 ) ( t 4 is given) t 1 t 2 t 3 t 4 t 5 t 6 t 7 q ( t ) p ( t ) Linear equations 3-5 Applications a set of linear equations Ax = b (with A m × n ) is • square...
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This note was uploaded on 10/15/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Spring '08 term at UCLA.

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lineqs - EE103 (Fall 2009-10) 3. Linear equations •...

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