|DISCLAIMER: To do this problem you need to know 1) How to invert a 3x3 matrix and 2) How to multiply matrices, I will not be showing my work on how I did that, as that would take literally forever.
That being said we are given:
T(from beta to beta) =
3 1 1
2 5 2
5 6 3
We are also given our friend's basis vectors, these three vectors we will combine into a matrix that we'll give an arbitrary name "Q". Another name for Q is the Identity matrix (from gamma to beta).
Q / [I] (from gamma to beta) =
1 2 2
0 3 1
2 2 0
If you remember from class, our homie David gave us the formula:
Q^-1 * [T](from beta to beta) * Q = [T](from gamma to gamma)
We are looking for [T](from gamma to gamma), so we'll be using this formula.
For this formula, the first thing we need to find is the inverse of Q. Another name for the inverse of Q is the Identity matrix (from beta to gamma)
Q^-1 / [I] (from beta to gamma) =
1/5 -2/5 2/5
-1/5 2/5 1/10
3/5 -1/5 -3/10
Now, the first step in the formula is to multiply the inverse of Q (Q^-1) by the linear transformation T ([T] from beta to beta) that we were given.
Q^-1 * [T](from beta to beta) =
9/5 3/5 3/5
7/10 12/5 9/10
-1/10 -11/5 -7/10
And the final step to find our answer ([T] from gamma to gamma) is to multiply the matrix we just solved for (Q^-1 * [T] from beta to beta) with our friend's basis Q.
Q^-1 * [T](from beta to beta) * Q =
3 33/5 21/5
5/2 52/5 19/5
-3/2 -41/5 -12/5
There's our final answer.