Transcription of Introduction to Linear Algebra, 5th Edition
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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.
This section starts with three vectors u,v,w. I will combine them using matrices. Three vectors u = 1 −1 0 v = 0 1 −1 w = 0 0 1 . Their linear combinations in three-dimensional space arex 1u+x 2v +x 3w: Combination of the vectors x 1 1 −1 0 + x 2 0 1 −1 + x 3 0 0 1 = x 1 x 2 −x 1 x 3 −x 2 . (1)
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