Ee1034aS10

Ee1034aS10 - A Review of Basic Linear Algebra EE103 SLIDES...

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A Review of Basic Linear Algebr A Review of Basic Linear Algebra EE103 SLIDES 4A (SEJ) 1 The Matrix Inverse, and its Non Use 11 1 12 2 1 1 nn ax ax ax b   ... ................................. 11 1 n n ax b .............. A x b 1 If has an inverse A xA b I Not for computation! f has an invers ! e EE103 SLIDES 4A (SEJ) 2
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The Matrix Inverse, and its Non Use 12 ( 1 , 0 ,, 0 ) ' , ( 0 , 1 , 0 0 ) ' ( 0 0 , 1 ) ' n ee e   1 i A xe i n , , ..., n xx x [, , . . . , ] 1 1 nn n nxn A x A x A x e e I [ , ,..., ] [ ] [ ] EE103 SLIDES 4A (SEJ) 3 Examples fxx x f n 112 0 0 ,,, b g x 1 2 2 2 3 2 9 1  x x x n 212 b g x x x x 123 1 2 3 2 0  x 0 b g 00 0 0 T        0 11 1 f x f x f x    , n x x       0 22 2 f x f x f x 0 n f x f x f x EE103 SLIDES 4A (SEJ) 4
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Examples      00 0 11 0 0 0 fx fx xx xx  0 0 (1, 2,1) ( ) (242 ) x       22 0 1 0 2 0 ()( 2 , 4, 2) 3 , 1 , 2 )      0 nn f xf x x x 0 () 3 1 , 1, 2) f x       0    1 2 24 2 1 3 31 2 2 1 x x   0 3 11 2 1 2 x   EE103 SLIDES 4A (SEJ) 5 0 y f J x yf x 1 2 3 1 y y 100 xxy  3 2 y T   0 1.2143, 0.9286, 0.8571 y   1 0.2143 , 2.9286, 1.8571 T x EE103 SLIDES 4A (SEJ) 6
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Examples: Polynomial Interpolation and Approximation [x y] = 1 1 2 -2 3 -4 43 4 3 5 -4 6 1 7 2 7 -2 8 9 EE103 SLIDES 4A (SEJ) 7 Polynomial Interpolation and Approximation () 1 bt i i th ,1 , , 1 ii xyi n m n  ( , ), ,, nobservations x y n degree algebraicpolynomial m  ? 11 1 1 0 1 ? m m m px ax ax a y y    22 1 2 0 2 m ? 10 m nm n n n y The variables are: , mm aa a EE103 SLIDES 4A (SEJ) 8
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Polynomial Interpolation and Approximation 1. If mn  1 , there are typically many solutions 2. If  1 , there generally will be no solution. 3. If  1 (a square system), there is, under nonsingularity, a unique solution. 12 11 1 1 1 nn n a y x xx   22 2 1 n ay x   0 1 n n x  EE103 SLIDES 4A (SEJ) 9 Polynomial Interpolation and Approximation ? ,1 , , 1 ii m xyi n m n px ax ax a y     1 1 0 1 ? 1 2 0 2 m m m y ? m y   10 nm n n n 1 variables n linear equations in m more rows than columns EE103 SLIDES 4A (SEJ) 10
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Least Squares : ,x , The latter is a linear system of the form Ax b A is m n and m n  Schematically       EE103 SLIDES 4A (SEJ) 11  Example: Linear Least Squares Ax b m n e x Ax b () | | ( ) | | ex Ax b Ax b t 2 2 m i n | | ( ) | | m i n x x t 2 2 Expanding the term on the right hand side, we have min 2 tt t t x xAAx bAx bb  EE103 SLIDES 4A (SEJ) 12
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Example: Linear Least Squares
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This note was uploaded on 06/01/2010 for the course EE EE 103 taught by Professor Jacobsen during the Spring '09 term at UCLA.

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Ee1034aS10 - A Review of Basic Linear Algebra EE103 SLIDES...

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