Matrix of Transformation (non standard basis)

[margaret37]

Homework Statement

Define T : R^3x1 to R^3x1 by T = (x1, x2,x3)^T = (x1, x1+x2, x1+x2+x3)^T

1 Show that T is a linear transformation
2 Find [T] the matrix of T relative to the standard basis.
3 Find the matrix [T]' relative to the basis
B' = {(1,0,0)^t, (1,1,0)^t, (1,1,1)^t
4 Find the transition matrix Q from B to B'
5 Verify that [T]' = Q[T]Q^-1

Homework Equations

The Attempt at a Solution

1) I think this is done I showed that T(c(alpha + beta)) = c* T(alpha) + T(Beta) by substituting in vectors (a1, a2, a3) and (b1,b2,b3) and working through the algebra.

2) This is what I did, I think I understood this properly:

I built a transformation Matrix [T] = T(e1) | T(e2) | T(e3)

I got
1 0 0
1 1 0
1 1 1

3) Then I think is wrong but I'm not sure

I built another matrix [T]' = T(v1) | T(v2) | t(v3)
where v1, v2, and v3 are the the vectors given above

I got
1 1 1
1 2 2
2 2 3


4) Then I built a transformation matrix by finding the coordinates of e1, e2, and e3 relative to v1, v2 and v3

I got
1 -1 0
0 1 -1
0 0 1

I calculated an inverse

1 1 0
0 1 1
0 0 1

5) Then if I had done everything right
[T]' should have been equal to Q[T]Q^-1

But it wasn't even close

1 1 1
0 1 1
0 0 1

Any help would be greatly appreciated.
Margaret

 

[Billy Bob]

3) Then I think is wrong but I'm not sure

I built another matrix [T]' = T(v1) | T(v2) | t(v3)
where v1, v2, and v3 are the the vectors given above

I got
1 1 1
1 2 2
2 2 3

One error is that T(v1) should be (1,1,1)^t instead of (1,1,2)^t. A more important error is that the resulting numbers "1,1,1" are coefficients with respect to the standard basis, not B'. You need to figure out x, y, and z such that
(1,1,1)^t=x v1 + y v2 + z v3. This is easy to do by inspection: x=0, y=0, z=1. The first column of [T]' is therefore (0,0,1)^t. Do a similar computation for T(v2) and T(v3). The appropriate coefficients x, y, and z are easy to find by inspection, although not as trivial as for T(v1).

Also, the inverse of

1 -1 0
0 1 -1
0 0 1

is almost but not quite

1 1 0
0 1 1
0 0 1

The row 1 column 3 entry is incorrect.

 

[margaret37]

Thank you very much. I think I've got it now.