Co-ordinate transformation matrix

[asif zaidi]

Plz advise if my approach is correct for 1st part and for 2nd part, I need some help.


Problem Statement
Consider the linear transformation T: R3->R2 whose matrix with respect to standard bases is given by | 2,1,6 |
| 0,2,-1|.
Now consider the bases f1={2,4,0}, f2={1,0,1}, f3 = {0,3,0} in R3 and
g1 = {1,1} and g2 = {1,-1}

Compute the coordinate transformation matrices between the standard bases and these bases and compute the matrix of T with respect to the new bases

Problem solution

For first part, I am doing the following
Express u1, u2, u3 in terms of standard bases vectors
u1 = 2e1 + 4e2;
u2 = e1 + e3;
u3 = 3e2;
Solve for e1,e2,e3 in terms of u1, u2, u3 and transpose of this is my co-ordinate transformation matrix. Is this correct?

For g vectors, do in a similar manner

For second part
I don't understand this part. How do I compute the matrix of T wrt these new bases. Any pointers would be appreciated.

 

[FunkyDwarf]

Look where T takes the new standard basis elements ie equiv of (1,0,0), (0,1,0) etc

 

[asif zaidi]

Don't really understand your response. Could you elaborate please.

Also, is my 1st part to solution correct?

Thanks

 

[radou]

I don't understand this part. How do I compute the matrix of T wrt these new bases. Any pointers would be appreciated.

Look at how exactly the matrix representation of an operator is given.