Chứng minh rằng một tự đồng cấu f : V → V là đẳng cự khi và chỉ khi nó bảo toàn tích vô hướng, tức là

hf(x)|f(y)i = hx|yi, ∀x, y ∈ V.

The distribution of the amount of money spent by students for textbooks in a semester is approximately normal in shape with a mean of $235 and a standard deviation of $20. according to the standard deviation rule, how much did almost all (99.7%) of the students spend on textbooks in a semester?

Elena is paid a constant rate for each hour she works. the table shows the amounts of money that elena earned for various amounts of time that she worked.

Aroulette wheel has 38 slots in which the ball can land. two of the slots are green, 18 are red, and 18 are black. the ball is equally likely to land in any slot. the roulette wheel is going to be spun twice, and the outcomes of the two spins are independent. the probability that it lands one time on red and another time on black in any order is(a) 0.5 (b) 0.25 (c) 0.4488 (d) 0.2244