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Introduction to Linear Algebra, 5th Edition

22 Chapter 1. Introduction to Matrices'&$%1A= 1 23 45 6 is a3by2matrix :m= 3rows andn= 1 23 45 6 x1x2 is acombination of the columnsAx=x1 135 +x2 246 .3 The3components ofAxare dot products of the3rows ofAwith the vectorx:Row at a time 1 23 45 6 78 = 1 7 + 2 83 7 + 4 85 7 + 6 8 = 235383 .4 Equations in matrix formAx=b: 2 53 7 x1x2 = b1b2 replaces2x1+ 5x2=b13x1+ 7x2= solution toAx=bcan be written asx=A 1b. But some matrices don t allowA section starts with three vectorsu,v,w. I will combine them vectorsu= 1 10 v= 01 1 w= 001 .Their Linear combinations in three-dimensional space arex1u+x2v+x3w:Combinationof the vectorsx1 1 10 +x2 01 1 +x3 001 = x1x2 x1x3 x2 .(1)Now something important:Rewrite that combination using a matrix. The vectorsu,v,wgo into the columns of the matrixA. That matrix multiplies the vector(x1, x2, x3) :Matrix times vectorCombination of columnsAx= 1 0 0 1 1 00 1 1 x1x2x3 = x1x2 x1x3 x2 .(2)The numbersx1, x2, x3are the components of a vectorx.

Introduction to Vectors ... That pattern would continue for a 4 by 4 difference matrix. The next square would be x 4 = 16. The next difference would be x 4 −x 3 = 16 −9 = 7 (the next odd number). The matrix finds all the differences 1,3,5,7 at once. Important Note: Multiplication a …

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