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Unformatted text preview: Lecture 10 Today’s class: • Review of matrix algebra • Matlab matrix operations • Solving linear systems of equations • Case Study: Global Positioning Systems (GPS) Matrices • Matrix [ A ] ∈ R p × q is rectangular array of numbers [ A ] = a 1 , 1 a 1 , 2 ··· a 1 ,q 1 a 1 ,q a 2 , 1 a 2 , 2 ··· a 2 ,q 1 a 2 ,q . . . . . . . . . . . . a p 1 , 1 a p 1 , 2 ··· a p 1 ,q 1 a p 1 ,q a p, 1 a p, 2 ··· a p,q 1 a p,q • Numbers a i,j = elements of [ A ] = entries of [ A ] • First index i of element a i,j = row index • Second index j of element a i,j = column index Vectors • pvector : “skinny” matrix (dimension p × 1 or 1 × p ) { x } = x 1 x 2 . . . x p 1 x p or { x } T = x 1 , x 2 , ··· , x p • Elements x i = components of { x } • Convention: braces denote column vectors i.e., assume { x } ∈ R p means { x } ∈ R p × 1 Vectors/matrices in Matlab • Columns separated by , (optional), rows by ; • Indexing using parentheses or colon notation: a 2 , 3 7→ A ( 2 , 3 ), x 3 7→ x ( 3 ) or x ( 3 , 1 ) or x ( 1 , 3 ) A (: , 2 ) means 2 nd column of A A ( 3 , :) means 3 rd row of A A ( 1 : 3 , 2 : 5 ) means rows 1 through 3 columns 2 through 5 of A A = [ 1 2 3; 4 5 6 ]; size(A), size(A,1), size(A,2) • Dimensions of matrices returned by function size Matrix operations • Matrix addition: given [ A ] and [ B ] conformable [ C ] = [ A ] + [ B ] ⇒ c ij = a ij + b ij • Scalar multiplication: given scalar α ∈ R , matrix [ A ], [ C ] = α [ A ] ⇒ c ij = αa ij • Matrix multiplication: given [ A ] ∈ R p × n , [ B ] ∈ R n × q , [ C ] = [ A ][ B ] ⇒ c ik = n X j =1 a ij b jk • Matrix transpose: given [ A ] ∈ R p × q , [ C ] = [ A ] T ⇒ c ij = a ji Zero and identity matrices • Zero matrix [0] ∈ R p × q is [0] = ··· ··· . . . . . . . . . ··· • [0] satisfies [ A ]+[0] = [0]+[ A ] = [ A ] for every [ A ] ∈ R p × q • Identity matrix I ∈ R p × p is [ I ] = 1 ··· 1 ··· ....
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This note was uploaded on 06/20/2011 for the course MATH 2070 taught by Professor Aruliahdhavidhe during the Winter '10 term at UOIT.
 Winter '10
 aruliahdhavidhe
 matlab, Systems Of Equations, Equations, Linear Systems, Matrices, Matrix Operations

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