Finding a Transformation Matrix

Date: 12/29/2003 at 12:08:49
From: Zdenko
Subject: base and transformation matrix

Vectors a1(4,2,1), a2(5,3,2), a3(3,2,1) and b1(-1,4,0), b2(4,3,1), b3
(-5,7,-3) are two bases for V3. What is the transformation matrix 
from first to second base?

The given result is:

 -5  0   4
 -4  -1  4
 13  3  -1

but I don't seem to be able to get that answer.  Can you help me?

Date: 12/31/2003 at 17:39:37
From: Doctor Fenton
Subject: Re: base and transformation matrix

Hi Zdenko,

Thanks for writing to Dr. Math.  Since both {a1,a2,a3} (which I will
call the "a-basis") and {b1,b2,b3} (the "b-basis") are bases for R^3,
any vector v in R^3 can be expressed uniquely as

  v = A1*a1 + A2*a2 + A3*a3     for scalars A1,A2,A3, 

and also as

  v = B1*b1 + B2*b2 + B3*b3     for scalars B1,B2,B3.

The change of basis matrix M from the a-basis to the b-basis must be a
3x3 matrix such that given the coefficients (A1,A2,A3), the product

   M [A1] = [B1]
     [A2]   [B2]
     [A3]   [B3]  .

If v = a1, the first vector in the a-basis, then 

  [A1]   [1]
  [A2] = [0]
  [A3]   [0]  ,

and since multiplying a matrix M by the column vector [1 0 0]^t
(transpose) on the right gives the first column of the matrix, the
first column of M must be the coefficients [B1 B2 B3]^t of the expansion

   a1 = B1*b1 + B2*b2 + B3*b3 .

The second and third columns of M would be found similarly, by finding
the expansions in the b-basis of a2 and a3. 

Your "answer" would say that

   a1 = -5*b1 -4*b2 + 13*b3 ,

or

  (4,2,1) = -5*(-1,4,0) -4*(4,3,1) + 13*(-5,7,-3) ,

and this is clearly NOT true.

However, it IS true that 

   b1 = -5*a1 -4*a2 + 13*a3 , 

since

  (-1,4,0) = -5*(4,2,1) -4*(5,3,2) + 13*(3,2,1) .

Your given "result" is actually the change of basis matrix from the b-
basis to the a-basis, which is the inverse of the matrix you wanted in
the original statement of the problem.    
 
If you have any questions or need more help, please write back and
show me what you have been able to do, and I will try to offer further 
suggestions.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/01/2004 at 12:48:15
From: Zdenko
Subject: Thank you (base and transformation matrix)

Thanks for the help!