# Find the Standard Matrix of T

1. Feb 2, 2013

### x.x586

1. The problem statement, all variables and given/known data

Let T be a linear transformation from R3 to R3. Suppose T transforms (1,1,0) ,(1,0,1) and (0,1,1) to (1,1,1) (0,1,3) and (3,4,0) respectively.

Find the standard matrix of T and determine whether T is one to one and if T is onto

2. Relevant equations

3. The attempt at a solution

I know T(x) =Ax=[T(e1) ,T(e2,) T(e3)]

I thought A would just be the matrix with columns (1,1,1) (0,1,3) and (3,4,0), but then I realized that
(1,1,0) ,(1,0,1) and (0,1,1) are not the standard basis vectors for R3

My book doesn't give any examples where we don't start with the standard basis vectors

Should I have started by taking a 3x3 matrix entries [x1,x2,x3;x4,x5,x6] and multiply that by a 3x3 matrix with entries [1,1,0;1,0,1;0,1,1] and set that equal to a matrix with entries [1,0,3;1,1,4;1,3,0] and then got a system of equations from there by multiplying the left side out. And then set up an augmented matrix and used row reduction to find corresponding entries for A?

2. Feb 2, 2013

### csopi

Using the linearity of T, you can calculate T(e1), T(e2) and T(e3):

T(e1)+T(e2)=(1,1,1)
T(e1)+T(e3)=(0,1,3)
T(e2)+T(e3)=(3,4,0)

After that, you can build up A.

3. Feb 2, 2013

### vela

Staff Emeritus
Yes, that'll work. You could also write ei as a linear combination of the given vectors, and then use the linearity of T to evaluate T(ei).