What is the correct method for transforming a matrix to a new basis?

[transgalactic]

my solution is in that link

http://img246.imageshack.us/my.php?image=img8267rf4kq1.jpg

i solved it using that method

i want to transform a matrix from a standart basis to basis B
A-out old matrix
X- the old basis
Y-the new basis
S-the transformation matrix
S^-1 -its inverse

N-resolt
i did
Y|X>>>I|S

S|I=I|S^-1

N=S*A*S^-1

the problem is with the solution that i was given

but the problem is that the solution that i was given is totally different

http://img352.imageshack.us/my.php?image=45865659to7.jpg

A-old matrix
B-new basis
N-answer

i noticed that he takes
B|A>>> I|N

thats totally different i can't see any logic in that
and he got a different answer from me
and he got full points for that

that gives me a great dought in my method

can you please tell were am i wrong??

 

[John Creighto]

I'm not sure what you wrote down but if

Y=TX

where:

T is a transformation from x to y
then the matrix is transformed as follows

A2=T A T^-1

The basis vectors which you are given can be thought of as a mapping of the basis vectors to the x y z coordinates. Thus your problem differs slight in that I believe T^-1 has columns which are the x, y and z coordinates of the basis vectors which you specified.