I have attached the question on the post, and am really struggling to get this right

MATH HELP .pdf

My teacher gave this solution but i dont get how T(1) = -1 and T(t) = 1?

I know how to get the standar matrix, i.e by applying T(e1) and T(e2) to the transformation and I get [0 1 -1 0]

Can someone please explain this solution to me, i am really unable to follow it

And can someone please explain how he gest L(1+t) = 1-t? and how L(1-t) = -1-t ?

Thank you.