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by: Mary Veum

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# Applied Linear Algebra MATH 2210

Mary Veum
UCONN
GPA 3.97

Thomas Roby

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COURSE
PROF.
Thomas Roby
TYPE
Class Notes
PAGES
6
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 6 page Class Notes was uploaded by Mary Veum on Thursday September 17, 2015. The Class Notes belongs to MATH 2210 at University of Connecticut taught by Thomas Roby in Fall. Since its upload, it has received 25 views. For similar materials see /class/205825/math-2210-university-of-connecticut in Mathematics (M) at University of Connecticut.

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Date Created: 09/17/15
21 Matrix Operations Matrix Notation Two ways to denote m x n matrix A In terms of the columns ofA In terms of the entries ofA all 611 01quot A ah ay am an amj amquot Main diagonal entries Zero matrix 0 0 0 0 0 0 0 0 0 0 THEOREM 1 LetA B and C be matrices ofthe same size and let r and s be scalars Then aABBA drABrArB bABCABC ersArAsA CA0 A f rsA rsA Matrix Multiplication Multiplying B and X transforms X into the vector BX In turn if we multipIyA and BX we transform Bx into ABX SoABX is the composition of two mappings De ne the product1B so thatABx ABX SupposeA is m x n and B is n Xp where 961 962 B b1 b2 bpandx xp Then BX x1b1x2b2 prp and ABXAx1b1x2b2 prp Ax1b1 Ax2b2 Aprp x1 xlAb1szb2prbp Abz xz xp Therefore and by de ning AB Ab1 Abz Abp we have ABx ABX 472 375 01 EXAMPLE Compute AB whereA Solution gtAB andB 273 677 4 72 Abz 3 75 3 0 1 77 2 26 77 74 2 724 26 6 77 Note thatAb1 is a linear combination ofthe columns ofA and Abz is a linear combination of the columns ofA Each column ofAB is a linear combination ofthe columns ofA using weights from the corresponding columns ofB EXAMPLE lfA is 4 x 3 and B is 3 x 2 then what are the sizes ofAB and BA Solution gtllt gtllt BAwould be gtllt gtllt which is lfAismxnandBisnxpthenABismgtltp RowColumn Rule for Computing AB alternate method The definition AB Ab1 Abz wAbp is good for theoretical work When A and B have small sizes the following method is more efficient when working by hand fAB is defined let ABy denote the entry in the ith row and jth column ofAB Then anblj 612sz ambnj an 612 am 143 2 73 2 3 6 EXAMPLE A B 0 1 ComputeAB if it is de ned 71 0 1 4 77 Solution Since A is 2 x 3 and B is 3 x 2 then AB is defined and AB is x 2 73 2 73 2 3 6 28 I 2 3 6 28 745 AB 0 1 0 1 71 0 1 I I 71 0 1 I I 4 77 4 77 2 73 2 73 2 3 6 28 745 2 3 6 28 745 0 1 0 1 71 0 1 2 I 71 0 1 2 4 4 77 4 77 28 745 SoAB 2 74 THEOREM 2 LetA be m x n and let B and C have sizes for which the indicated sums and products are defined a ABC ABC associative law of multiplication b AB C AB AC left distributive law c B CA BA CA rightdistributive law d rAB rAB ArB for any scalar r e MA A A1quot identity for matrix multiplication WARNINGS Properties above are analogous to properties of real numbers But NOT ALL real number properties correspond to matrix properties 1 It is not the case thatAB always equal BA see Example 7 page 114 2 Even ifAB AC then B may not equal C see Exercise 10 page 116 3 It is possible forAB 0 even ifA 0 and B 0 see Exercise 12 page 116 Powers ofA Ak AA EXAMPLE liilliilliilliill lliillziil lfA is m x n the transpose ofA is the n x m matrix denoted byAT whose columns are formed from the corresponding rows ofA EXAMPLE 167 12345 276 A 67898 2 AT385 76543 494 583 1 2 1 2 0 EXAMPLE LetA 3 0 1 B 0 1 ComputeAB ABTATBTand BTAT Solution 120 12 AB 01 3 01 724 ABT 13 10 2 7 310 ATBT 20 2074 214 01 214 10 2 13 BTAT 7 1 1 214 01 THEOREM3 LetA and B denote matrices whose sizes are appropriate for the following sums and products a ATT A Le the transpose ofAT is A b A BT ATBT c For any scalar r rAT rAT d ABT BTAT le the transpose ofa product of matrices equals the product oftheir transposes in reverse order EXAMPLE Prove that ABCT Solution By Theorem 3d ABCV ABCT CT T CT

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