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posted by  bjjad on 8/23/2009 11:04:32 AM  |  status: Closed  |  Earned Karma: 50

Linear Transformations

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Linear Algebra Linear Algebra by J.B. Fraleigh and R.A. Beauregard (any edition), Addison Wesley; N/A N/A N/A
Question Details:

In each of the following assume that T is a linear transformation.
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Oracle
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(andhra university college of engg.)
posted by VENUGOPAL on 8/23/2009 12:08:56 PM  |  status: Live
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Response Details:
ANSWERED ALREADY
SEE BELOW
If T([1, 1, 0]) = [2, 5; 1; 3], T([1; 2; 0]) = [-1; 1; 2; 2] and T([1; 2; 1]) =[1; 0; 0; 2],
find the standard matrix of T and use it to determine T([1; 1; 1]).
THIS IS A TRANSFORMATION FROM U IN  R3 TO V IN  R4
 THE BASIS B1 OF U CONSISTS OF  BASE VECTORS
B1=[U1,U2,U3] WHERE
U1=[1,1,0].....U2=[1,2,0]........U3=[1,2,1]......
CORRESPONDING BASIS B2 OF V CONSISTS OF  BASE VECTORS
B2=[V1,V2,V3] WHERE
V1=[2; 5; 1; 3]........... V2=[-1; 1; 2; 2]........V3=[1; 0; 0; 2],
HENCE THE TRANSFORM MATRIX A  OF T[B1 TO B2] IS GIVEN BY

TO FIND T[1,1,1] LET US WRITE
U=[1,1,1] AS A LINEAR COMBINATION OF U1,U2,U3

Z=1
X+2Y+2Z=1........X+2Y=-1
X+Y+Z=1............X+Y=0
Y=-1....X=1
THAT IS [1,1,1]=1U1-1U2+1U3
HENCE
T[1,1,1]=1*T[U1]-1*T[U2]+1*T[U3]=V1-V2+V3
=[2; 5; 1; 3]-[-1; 1; 2; 2]+[1; 0; 0; 2] =  [4,4,-1,3]
T[1,1,1] = [4,4,-1,3]



venugopal.a.
Oracle
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posted by RedSox123 on 8/23/2009 11:04:40 PM  |  status: Live
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Response Details:
We want to express [1, 1, 1] in terms of [1, 1, 0], [1, 2, 0], and [1, 2, 1].

This is a system of 3 equations.
[1, 1, 1] = x*[1, 1, 0] + y*[1, 2, 0] + z*[1, 2, 1]

Solving for x, y, z, we get... 
x=1
y=-1
z=1

Since the transform is linear, we know
T(a + b) = T(a) + T(b)

Or in this case
T([1, 1, 1]) = T([1, 1, 0] - [1, 2, 0] + [1, 2, 1]) = 
T([1, 1, 0]) - T([1, 2, 0]) + T([1, 2, 1])

Now, just plug in your values for each of those and simplify.
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