How Do I Find the Transition Matrix Between Matrices C and B?

[robertjford80]

Homework Statement

I'm enormously frustrated with these problems. I've been trying to figure out how to find out what the transition matrix between C and B is for about 2 hours and I still can't get it. I've watched 4 youtube videos and read two websites as well as the section in my textbook. I still can't get it. Anyway, this one video on youtube said that to get the transition matrix between these two matrices

[1 -1 0] [-1 1 0]
[0 0 1] [1 2 1]
[1 0 2] [0 -1 0]

you just put them together in a 3 by 6 matrix and reduce it to reduced row echelon form and the 3 by 3 matrix on the right is your transition matrix in this case

[-2 -5 -2]
[-1 -6 -2]
[1 2 1]

Well that method doesn't work for the problems I'm working on. Number 17 and 21. (I can get 19)

The Attempt at a Solution

Using this calculator (if I calculated everything by hand I would have wasted 5 hours already)
http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=roc

I get

[5 4 6]
[4 3 4]
[-2 -2 3]

the book says the answer is

[-7 -6 12]
[6 5 -10]
[-4 -4 7]

I can't get the right answer for 17 either but we'll just worry about 21 for now.

 

[robertjford80]

I'm still open to help for this problem, if anyone is capable.

 

[Fightfish]

Lets look at problem 21, for instance.
Observe that:
$\left( \begin{array}{c} -1\\ 2\\ 1 \end{array} \right) = -7 \left( \begin{array}{c} 1\\ 0\\ -1 \end{array} \right) + 6 \left( \begin{array}{c} 1\\ 1\\ 1 \end{array} \right) - 4 \left( \begin{array}{c} 0\\ 1\\ 3 \end{array} \right)$
So, the first column of my transition matrix will be $\left( \begin{array}{c} -7\\ 6\\ -4 \end{array} \right)$
Decomposing the other two B basis vectors in terms of the C basis vectors in similar fashion will yield the two other columns.