A First Course in Linear Algebra by Robert A Beezer PDF

Linear

By Robert A Beezer

ISBN-10: 0984417559

ISBN-13: 9780984417551

A primary path in Linear Algebra is an creation to the elemental options of linear algebra, in addition to an advent to the ideas of formal arithmetic. It starts off with platforms of equations and matrix algebra ahead of stepping into the idea of summary vector areas, eigenvalues, linear modifications and matrix representations. It has a number of labored examples and routines, besides certain statements of definitions and entire proofs of each theorem, making it excellent for self sustaining research.

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Together this means that d1 = d1 . Second Step. Suppose that we have determined that d1 = d1 , d2 = d2 , d3 = d3 , . . dp = dp . Let’s now show that dp+1 = dp+1 . Working towards a contradiction, suppose that dp+1 < dp+1 . For 1 ≤ ≤ p, 0 = [B]p+1,d m = δp+1,k [C]kd k=1 Definition RREF §R R E F Beezer: A First Course in Linear Algebra 34 m = δp+1,k [C]kd k=1 m = δp+1, [C] + d δp+1,k [C]kd Property CACN k=1 k= m δp+1,k (0) = δp+1, (1) + Definition RREF k=1 k= = δp+1, Now, Definition RREF 1 = [B]p+1,dp+1 m δp+1,k [C]kdp+1 = k=1 p m = δp+1,k [C]kdp+1 δp+1,k [C]kdp+1 + k=1 p = Property AACN k=p+1 m δp+1,k [C]kdp+1 (0) [C]kdp+1 + k=p+1 k=1 m = δp+1,k [C]kdp+1 k=p+1 m = δp+1,k (0) dp+1 < dp+1 k=p+1 =0 This contradiction shows that dp+1 ≥ dp+1 .

This will help you count, and identify, the pivot columns. In your work, you can box ’em, circle ’em or write ’em in a different color — just identify ’em somehow. This device will prove very useful later and is a very good habit to start developing right now. Example SAB Solutions for Archetype B Let’s find the solutions to the following system of equations, −7x1 − 6x2 − 12x3 = −33 5x1 + 5x2 + 7x3 = 24 x1 + 4x3 = 5 First, form the augmented matrix, −7 5 1 −6 5 0 −12 −33 7 24 4 5 and work to reduced row-echelon form, first with j = 1, 1 0 4 5 1 −5R1 +R2 5 5 7 24 −−−− −−→ 0 −7 −6 −12 −33 −7   1 0 4 5 7R1 +R3 −−− −−→  0 5 −13 −1 0 −6 16 2 0 5 −6 R ↔R 3 −−1−−−→ 4 5 −13 −1 −12 −33 Now, with j = 2,  1 −−−→  0 0 1 5 R2 And finally, with j = 3,  1 5 R 3 2 −− −→  0 0  1 −4R3 +R1 −−−− −−→  0 0 0 1 −6 0 1 0 0 1 0 4 −13 5 16 4 −13 5 1 0 0 1  6R2 +R3  13 5 R3 +R2  1 −1  −−−−−→  0 5 2 0 5 5  1 −1  −−−−−−→  0 5 0 2  −3 5 2 0 1 0 0 1 0 4 −13 5 2 5 5  −1  5 4 5  4 5 0 5 1 2 This is now the augmented matrix of a very simple system of equations, namely §R R E F Beezer: A First Course in Linear Algebra 38 x1 = −3, x2 = 5, x3 = 2, which has an obvious solution.

Theorem REMEF Row-Equivalent Matrix in Echelon Form Suppose A is a matrix. Then there is a matrix B so that §R R E F Beezer: A First Course in Linear Algebra 30 1. A and B are row-equivalent. 2. B is in reduced row-echelon form. Proof. Suppose that A has m rows and n columns. We will describe a process for converting A into B via row operations. This procedure is known as Gauss-Jordan elimination. Tracing through this procedure will be easier if you recognize that i refers to a row that is being converted, j refers to a column that is being converted, and r keeps track of the number of nonzero rows.

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