Mystery Minus Sign - Be a Detective! :) (Can't figure out after 2 hours)

I've spent the past couple hours trying to figure out why this matrix multiplication isn't coming out the way it should.

The problem statement and my work is attached. Basically, I need to verify the theorem that says:

[UT]=[T]*[U]

However, when I multiply [T]*[U], a minus sign gets switched around (see pic).
Also, the bases I'm supposed to use are written on my work (they are different from those in the problem statement).

If anyone could find out what's wrong I'd appreciate it (my brain will too). I'll click the thank you button for you too :)

first of all, it should be [UT] = [U]*[T], yes?

secondly, please attempt to write your problem out in your question. attached images of photos are very hard to read, and a lot of posters here won't even bother opening attachments. there is an entire forum section devoted to LaTex and how to use it, you should at the very least give the tutorial thread a look-see.

without going into the extraordinate amount of detail required to check all your work, i can only indicate that i suspect your matrices are wrong, because your computation of T is wrong.

if T(f(x)) = f'(x)(3+x) + 2f(x), then T(1-x) = (-1)(3+x) + 2(1-x) = -3 - 3x + 2 - 2x = -5x - 1, not -1 - 3x as you have.

I was actually wondering about that. I do apologize, especially for this one it was hard to read. I'll look into using latex next time. Thanks again for the help!

Hmmm I still think it equals -1-3x. I'm not sure how you'd get -5x-1 (Doh)

my calculations were shown in my previous post, for all to see. it's possible i may have made an error, because i do that sometimes, but if so, it is there to check.