Let L : V --,I- IV be a linear tmnsfonnation. and let dim V = dim IV. Prove that L is invertible if and only if [he image of a basis for V under L is a basis for IV.

Bayes Theorem Bayes Theorem Bayes Theorem P | A (A) If A and T are events, then: P |T P | AP(A) P T| A' (A') The events A and A’ form a partition of the sample space. This means • their union is the whole sample space and • the intersection of the two events A and A’ is empty. Bayes Theorem Given: P | B 0.3 P( ) 0.9 P(B’) = 1-0.9 P(A| B') 0.5 Find: P | A P B | A P | B (B) PA| B (B) P A|B' P(') PB | A 0.3(0.9) .84375 PB | A 0.8438 0.3(0.9) 0.5(0.1) Bayes Theorem Suppose that it snows in Greenland an average of once every 27 days, and when it does, glaciers have a 21% it snows (S) in chance of growing. When it does not Greenland an average snow in Greenland, glaciers have only a of once every 27 days, 6% chance of growing. What is the … probability that it is snowing in When it does not snow Greenland when glaciers are growing (S’) in Greenland, P G |S P(S) P |G PG |S (S) P G|S ' PS ') / 27(0.21) P |G / 27(0.21) 26/ 27(0.06) PS |G 0.1186 Bayes Theorem In a certain year, 58% of all Caucasians in the U.S., 52% of all African-Americans, 55% of all Hispanics, and 58% of residents not classified into one of these groups u