Define the linear transformation T by T(x) Ax. Find ker(T) , nullity(T) , range(T), and rank(T).a)ker(T)(b)nullity(T)(c) range(T) {(~t,t): t is any real number}R2{(...

Let $T: P_{2}(\mathbb{R}) \rightarrow P_{2}(\mathbb{R})$ be the linear transformation satisfying $$T(1)=x+1, \quad T(x)=x^{2}-1, \quad T\left(x^{2}\right)=3 x+2$$. Determine $T\left(a x^{2}+b x+c\right),$ where $a, b,$ and $c$ are arbitrary real numbers.

So that's all this problem. First note that t of two x equals four x And because tea is linear, I can move my constant coefficient outside so I end up with two times. TLX is equal to four x, which implies that t of axe equals two acts after dividing both sides by two. Secondly, if we look at t of three X Plus two, that's gonna be equal to two x plus three. But T is linear. That means I can distribute my teeth overthrew the individual terms. That means we end up with t 03 x plus t of to and just being the two on the right side, we get two x Plus six similar. Easy as before. Tea is linear is like move the cost a coefficient outside of tea. So now we get three times to you vax plus two times t of one because two times one is equal to which is equal to two experts. Six. Now we actually know that TX is actually go to act so well. Something to that in and we'll have six x plus two times t of one is equal to two x plus six It's a fact ing things moving things across we get to attempt to you. One is equal to six minus four x drying everything by two. We end up with TF. One is equal to three minus two x. So now if we start with, uh, let's see t a x square minus one, which is equal to X squared plus X minus three teas Linear. That means we could distribute t to the individual terms. So we we get end up getting t of X square minus two. You have one, but we really solved for what two of one is So in the next line, we can substitute to you if one for three minus two x So we end up with Kiev X square minus three minus two acts which, of course, is still equal to X squared plus X minus three being TX word by itself. We end up with the expression X squared minus X. So the salt the last part of expression If we start with key of a X square, plus BX plus C right, we know we can distribute because two years later we could distribute T to the individual terms so That means we end up with t of a X Square Krusty of Be Axe Plus to your seat. But T is linear so we can move the constant terms back throughout the concert terms outside of the of our transformations. So then we end up with a t of X square, plus b times t of X plus c times t of one because one time C is equal to see. So now we can substitute what we know what we know TLX Square to have accent here one are equal to and what we end up with is a times X squared minus x both the times to EPS plus C time three minus two acts. So if we distribute everything ah, and rearrange stuff, you end up with a X square minus a minus to B plus to see times X plus three seat