A formal power series over R is a formal infinite sum f = X[infinity] n=0 anxn, where the coefficients an ∈ R. We add power seri
es term-by-term, and two power series are the same if all their coefficients are the same. (We don’t plug numbers in for x, because we don’t want to worry about issues with convergence of the sum.) There is a vector space V whose elements are the formal power series over R. There is a derivative operator D ∈ L(V ) defined by taking the derivative term-by-term: D X[infinity] n=0 anxn ! = X[infinity] n=0 (n + 1)an+1xn What are the eigenvalues of D? For each eigenvalue λ, give a basis of the eigenspace E(D, λ). (Hint: construct eigenvectors by solving the equation Df = λf term-by-term.)
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Answer:
Check the explanation
Step-by-step explanation:
where the letter D is the diagonal matrix with diagonal entries λ1,…,λn. Now let's assume V is invertible, that is, this particular given eigenvectors are linearly independent, you get M=VDV−1.
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Answer:
The probability is
Step-by-step explanation:
The game of roulette wheel consists in spinning a wheel with 38 slots : 18 red, 18 black and 2 green.
If we suppose that the roulette is a fair roulette, then each slot has an equal chance of capturing the ball and given that the ball lands in a red, black or green slot this event doesn't give us information about the following spin. Meaning that exists independence between each spin of the roulette wheel.
In our exercise, we spin the roulette wheel 3 consecutive times and each time the ball lands in a red slot. This doesn't give us information about the fourth spin (because of the independence).
Given that each slot has an equal chance of capturing the ball we can calculate the probability of the ball landing on a red slot on the next spin as :
We divide the favourable cases (18 red slots) by the total cases (38 slots) in order to calculate the probability.
The probability is