For each linear operator T on V, find the minimal polynomial of T. (a) V = R2 and T(o, b) = (a + b, a - b) (b) V = P2(R) and T(g(x)) = g'(x) + 2g(x) (c) V = P2(R) and T(f(x)) = -xf"(x) + f'(x) + 2f(x) (d) V = MnXn(i?) and T(A) = A*. Hint: Note that T2 = I.

Stat notes week 14 Hypothesis testing on the calculator - STAT, TEST, 1-PropZTest - p = p (null proportion/hypothesis) o - x= how many got right - n= how many times tried - Prop: = / (not equal to), > (significantly above), < (significantly below) - Ex. Mr. Baugh claims to be psychic and can guess the roll of a die. He threw 20 dice and picked 7 correctly. - p0= ⅙ because that is the chance he has if he is not psychic - x=7 because he picked 7 correctly - n=20 because he threw 20 times - Prop: > p0 Prequal info - Central limit theorem- if you add many independent random variables the distance approaches normal distributions Sampling distributions -