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Parametric Vector Solutions

Parametric Vector Solutions. p m shankar October 2014. A Homogeneous System Ax=0. b. ------- A ---------------. 1 -4 -2 0 3 -5 0 0 0 1 0 0 -1 0 0 0 0 0 1 -4 0.

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Parametric Vector Solutions

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  1. Parametric Vector Solutions p m shankar October 2014

  2. A Homogeneous System Ax=0 b ------- A --------------- 1 -4 -2 0 3 -5 0 0 0 1 0 0 -1 0 0 0 0 0 1 -4 0 1 -4 0 0 0 5 0 0 0 1 0 0 -1 0 0 0 0 0 1 -4 0 rref ([ A b]) Column #(s) [1 3 5] with "Pivots" --> BASIC variable(s). Column(s) [2 4 6] without pivots -->FREE variable(s) x1:----->>BASIC x2:--------->>FREE x3:----->>BASIC x4:--------->>FREE x5:----->>BASIC x6:--------->>FREE

  3. Solutions x1 = 0+4*x2-5*x6 =====> x2 = x2 x3 = 0+x6 =====> x4 = x4 x5 = 0+4*x6 =====> x6 = x6 x p v2 v4 v6 ------------------------------------------------------------------------ x1 0 4 0 -5 =====> x2 0 1 0 0 x3 = 0 + x2 0 + x4 0 + x6 1 =====> x4 0 0 1 0 x5 0 0 0 4 =====> x6 0 0 0 1 ------------------------------------------------------------------------ A*p-b --> must be a Null vector 0 0 0 A*v --> must be a Null Matrix 0 0 0 0 0 0 0 0 0

  4. A non-homogeneous system Ax=b (b is not NULL) b ------- A --------------- 1 0 1 0 20 0 1 -1 -1 0 1 1 0 0 80 0 0 0 1 60 1 0 1 0 20 0 1 -1 0 60 0 0 0 1 60 0 0 0 0 0 rref ([ A b]) Column #(s) [1 2 4] with "Pivots" --> BASIC variable(s). Column(s) [3] corresponds - -> FREE variable x1:----->>BASIC x2:----->>BASIC x3:--------->>FREE x4:----->>BASIC

  5. solutions x1 = 20-x3 x2 = 60+x3 =====> x3 = x3 x4 = 60 x = p v3 x1 20 -1 x2 = 60 + x3 1 =====> x3 0 1 x4 60 0 A*p-b --> must be a Null vector 0 0 0 0 A*v --> must be a Null vector 0 0 0 0

  6. A non-homogeneous systemNo FREE Variables (i. e., no v-vector) ----- A ----- b 1 1 2 8 -1 -2 3 1 3 -7 4 0 1 0 0 34/26 0 1 0 51/26 0 0 1 57/26 rref ([ A b]) There is a PIVOT in every Column. Basic variables only x1, x2 and x3 x p X1 X2 x3 34/26 51/26 57/26 =

  7. Matrix Inverse 1 0 -2 -3 1 4 2 -3 4 8 3 1 10 4 1 7/2 3/2 1/2 det(A)=2 A-1 directly from Matlab input A I 1 0 -2 1 0 0 -3 1 4 0 1 0 2 -3 4 0 0 1 Concatenated Matrix [A I] 1 0 0 8 3 1 0 1 0 10 4 1 0 0 1 7/2 3/2 1/2 RREF I A-1

  8. Matrix Inverse 1 -2 1 4 -7 3 -2 6 -4 det(A)=0 A-1 does not exist input A I 1 -2 1 1 0 0 4 -7 3 0 1 0 -2 6 -4 0 0 1 Concatenated Matrix [A I] 1 0 -1 0 3/5 7/10 0 1 -1 0 1/5 2/5 0 0 0 1 -1/5 1/10 RREF A-1 does not exist No identity Matrix

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