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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.

Introduction to Vectors ... linear algebra, and the output Ax is a linear combination of the columns of A. With numbers, you can multiply Ax by rows. With letters, columns are the good way. Chapter 2 will repeat these rules of matrix multiplication, and explain the ideas.

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Introduction to Linear Algebra, 5th Edition

math.mit.edu

1.3 Matrices 1 A = ... Linear Equations One more change in viewpoint is crucial. Up to now, the numbers x 1,x 2,x 3 were known. The right hand side b was not known. We found that vector of differences by multiplying A times x. Now we think of b as known and we look for x.

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