How Do You Find the Standard Matrix of a Linear Transformation?

[x.x586]

Homework Statement

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

Homework Equations

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?

 

[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.

 

[vela]

Homework Statement

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

Homework Equations

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?

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