Change of Basis- Please help

(Headbang)

Please help me, i just dont get change of basis. The book omits the steps and i dont understand the notation to understand what exactly i am supposed to do.

This is the example in the book, i dont understand how they arrive at the solution because i dont understand what the notation means and it is driving me insane already because i am sure that the actual calculations are easy.

x1+x2+x3=0 is a plane, consider the bases U and B in this plane.

U= (a1,a2)= (0,1,-1), (1,0,-1)

B= (b1,b2) = ( 1,2,-3),(4,-1,-3)

Find the change of basis matrix.

The book then does

[b1]subscript U= [2,1]

Also if I have diagnolized a matrix how do i find the change of basis matrix to go from the original to the diagnol?