Matrix representation after change of basis

Question

I am trying to solve this task using Wolfram Notebook. I would like to know how new matrix is calculated (part b) after a change of basis.

Part a: I decided to show this part because part b is a continuation of part a

Basis for y=-x:

p := {{ 0, -1 },{-1, 0}}

30 degrees rotation:

RotationTransform[30*Degree, {0, 0}][{x, y}]

q is 30 degrees rotation as matrix:

q:={{Sqrt[3]/2, -2^(-1)}, {1/2, Sqrt[3]/2}}

a:=p.q

Part b

New basis:

w:={{1,-1},{-1,2}}

I did not understand how exactly change of basis works, however, I tried to implement what I learned in this video:

Inverse[w].p.w.a

This way seems to me not correct.

Question: How can I calculate new matrix representation after a change of basis?

Answer 1

New matrix representation after a change of basis:

Inverse[w].a.w