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Apprentice
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posted by  agado88 on 8/25/2008 12:05:01 PM  |  status: Live  

linear algebra

Course Textbook Chapter Problem
Linear Algebra N/A N/A N/A
Question Details:
let z = [2]  ,  v1= [2]  ,  v2 = [5]
          [4]            [0]            [-2]  
          [0]            [-1]           [4] 
          [-1]           [-3]          [2]
 
Find the best approximation to z from the set of vectors in span {v1,v2}.
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Oracle
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posted by VENUGOPAL on 8/25/2008 1:26:46 PM  |  status: Live
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Response Details:
Question:
let z = [2]  ,  v1= [2]  ,  v2 = [5]
          [4]            [0]            [-2]  
          [0]            [-1]           [4] 
          [-1]           [-3]          [2]
 
Find the best approximation to z from the set of vectors in span {v1,v2}.
Z=AV1+BV2
2=2A+5B.....................1
4=-2B............................2
0=-A+4B.........................3
-1=-3A+2B.........................4
FOR THE BEST APPROXIMATION OF A AND B VALUES . WE SHOULD HAVE
S=[2A+5B-2]^2+(2B+4)^2+(A-4B)^2+(3A-2B-1)^2 SHOULD BE MINIMUM .
HENCE

2*2(2A+5B-2)+2(A-4B)+2*3(3A-2B-1)=0
8A+20B-8+2A-8B+18A-12B-6=0
28A-14=0
A=1/2 = 0.5......................1
2*5(2A+5B-2)+2*2(2B+4)+2(-4)(A-4B)+2*(-2)(3A-2B-1)=0
20A+50B-20+8B+16-8A+32B-12A+8B+4=0
98B=0
B=0
HENCE
Z=0.5V1 IS THE BEST APPROXIMATION.
PLEASE CHECK THE ARITHMETIC

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