MathsGee Homework Help Q&A - Recent questions and answers
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Powered by Question2AnswerSay you have \(k\) linear algebraic equations in \(n\) variables; in matrix form we write \(A X=Y\). Give a proof or counterexample for each of the following.
https://mathsgee.com/36512/algebraic-equations-variables-counterexample-following
Say you have \(k\) linear algebraic equations in \(n\) variables; in matrix form we write \(A X=Y\). Give a proof or counterexample for each of the following.<br />
<br />
a) If \(n=k\) there is always at most one solution.<br />
b) If \(n>k\) you can always solve \(A X=Y\).<br />
1<br />
c) If \(n>k\) the homogeneous equation \(A X=0\) has at least one solution \(X \neq 0\)..<br />
d) If \(n<k\) then for some \(Y\) there is no solution of \(A X=Y\).<br />
e) If \(n<k\) the only solution of \(A X=0\) is \(X=0\).Mathematicshttps://mathsgee.com/36512/algebraic-equations-variables-counterexample-followingFri, 21 Jan 2022 10:20:35 +0000Solve the given system - or show that no solution exists: \[ \begin{aligned} x+2 y &=& 1 \\ 3 x+2 y+& 4 z=& 7 \\ -2 x+y-& 2 z=&-1 \end{aligned} \]
https://mathsgee.com/36511/solve-given-system-show-solution-exists-begin-aligned-aligned
Solve the given system - or show that no solution exists:<br />
\[<br />
\begin{aligned}<br />
x+2 y &=& 1 \\<br />
3 x+2 y+& 4 z=& 7 \\<br />
-2 x+y-& 2 z=&-1<br />
\end{aligned}<br />
\]Mathematicshttps://mathsgee.com/36511/solve-given-system-show-solution-exists-begin-aligned-alignedFri, 21 Jan 2022 10:19:25 +0000Consider the system of equations \[ \begin{aligned} &x+y-z=a \\ &x-y+2 z=b . \end{aligned} \]
https://mathsgee.com/36510/consider-the-system-of-equations-begin-aligned-end-aligned
Consider the system of equations<br />
\[<br />
\begin{aligned}<br />
&x+y-z=a \\<br />
&x-y+2 z=b .<br />
\end{aligned}<br />
\]<br />
<br />
a) Find the general solution of the homogeneous equation, so \(a=b=0\).<br />
<br />
b) A particular solution of the inhomogeneous equations when \(a=1\) and \(b=2\) is \(x=1, y=1, z=1\). Find the most general solution of the inhomogeneous equations.<br />
<br />
c) Find some particular solution of the inhomogeneous equations when \(a=-1\) and \(b=-2\).<br />
<br />
d) Find some particular solution of the inhomogeneous equations when \(a=3\) and \(b=6\).<br />
<br />
[Remark: After you have done part a), it is possible immediately to write the solutions to the remaining parts.]Mathematicshttps://mathsgee.com/36510/consider-the-system-of-equations-begin-aligned-end-alignedFri, 21 Jan 2022 10:16:38 +0000If \(A\) is a \(5 \times 5\) matrix with \(\operatorname{det} A=-1\), compute \(\operatorname{det}(-2 A)\).
https://mathsgee.com/36509/times-matrix-with-operatorname-det-compute-operatorname-det
If \(A\) is a \(5 \times 5\) matrix with \(\operatorname{det} A=-1\), compute \(\operatorname{det}(-2 A)\).Mathematicshttps://mathsgee.com/36509/times-matrix-with-operatorname-det-compute-operatorname-detFri, 21 Jan 2022 10:15:32 +0000What was the initial distribution of the milk?
https://mathsgee.com/36508/what-was-the-initial-distribution-of-the-milk
Snow White distributed 21 liters of milk among the seven dwarfs. The first dwarf then distributed the contents of his pail evenly to the pails of other six dwarfs. Then the second did the same, and so on. After the seventh dwarf distributed the contents of his pail evenly to the other six dwarfs, it was found that each dwarf had exactly as much milk in his pail as at the start.<br />
What was the initial distribution of the milk?Mathematicshttps://mathsgee.com/36508/what-was-the-initial-distribution-of-the-milkFri, 21 Jan 2022 10:12:58 +0000Three friends sit around a table, each with a large plate of cheese. Instead of eating it, every minute each of them simultaneously pass half of their cheese to the neighbor on the left and the other half to the neighbor on the right.
https://mathsgee.com/36507/friends-around-instead-simultaneously-neighbor-neighbor
Three friends sit around a table, each with a large plate of cheese. Instead of eating it, every minute each of them simultaneously pass half of their cheese to the neighbor on the left and the other half to the neighbor on the right.<br />
a) Is it true that the amount of cheese on the first person's plate will converge to some limit as time goes to infinity? Explain.<br />
b) The next week they meet again, adding a fourth friend and follow the same procedure. What can you say about the eventual distribution of the cheese?Mathematicshttps://mathsgee.com/36507/friends-around-instead-simultaneously-neighbor-neighborFri, 21 Jan 2022 10:11:37 +0000Let \(T\) be the transition matrix for a Markov chain. If there is some power \(k\) for which all of the elements of \(T^{k}\) are positive, show that all the elements of \(T^{k+1}\) are positive.
https://mathsgee.com/36506/transition-matrix-markov-elements-positive-elements-positive
Let \(T\) be the transition matrix for a Markov chain. If there is some power \(k\) for which all of the elements of \(T^{k}\) are positive, show that all the elements of \(T^{k+1}\) are positive.Mathematicshttps://mathsgee.com/36506/transition-matrix-markov-elements-positive-elements-positiveFri, 21 Jan 2022 10:10:39 +0000Show that the matrix \(\lim _{k \rightarrow \infty} T^{k}\) has all of its columns equal to \(P_{\infty}\).
https://mathsgee.com/36505/show-that-matrix-rightarrow-infty-has-all-columns-equal-infty
If \(T\) is the transition matrix of a regular Markov process (so for some \(k\) all the entries of \(T^{k}\) are positive), we know there is a probability vector \(P_{\infty}\) so that if \(P_{0}\) is any initial probability vector, then \(\lim _{k \rightarrow \infty} T^{k} P_{0}=P_{\infty}\).<br />
Show that the matrix \(\lim _{k \rightarrow \infty} T^{k}\) has all of its columns equal to \(P_{\infty}\).Mathematicshttps://mathsgee.com/36505/show-that-matrix-rightarrow-infty-has-all-columns-equal-inftyFri, 21 Jan 2022 10:08:28 +0000Form a Markov chain by following the job of a randomly chosen child of a given family through several generations. Set up the matrix of transition probabilities.
https://mathsgee.com/36504/following-randomly-generations-transition-probabilities
Assume (naively) that a person's job can be classified as professional, skilled, or unskilled. Assume that, of the children of professional parents, 80 percent are professional, 10 percent are skilled, and 10 percent are unskilled. In the case of children of skilled, 60 percent are skilled, 20 percent are professional, and 20 percent are unskilled. Finally, in the case of unskilled, 50 percent of the children are unskilled, and 25 percent each are in the other two categories. Assume that every family has at least one child.<br />
a) Form a Markov chain by following the job of a randomly chosen child of a given family through several generations. Set up the matrix of transition probabilities.<br />
b) Find the probability that a randomly chosen grandchild of an unskilled worker is a professional.Mathematicshttps://mathsgee.com/36504/following-randomly-generations-transition-probabilitiesFri, 21 Jan 2022 10:07:32 +0000A certain plant species has either red (dominant), pink (hybred), or white (recesive) flowers, depending on its genotype.
https://mathsgee.com/36503/certain-species-dominant-recesive-flowers-depending-genotype
A certain plant species has either red (dominant), pink (hybred), or white (recesive) flowers, depending on its genotype. If you cross a pink plant with any other plant, the probability distribution of the offsprings are prescribed by the transition matrix<br />
\[<br />
T:=\left(\begin{array}{ccc}<br />
.5 & .25 & 0 \\<br />
.5 & .5 & .5 \\<br />
0 & .25 & .5<br />
\end{array}\right)<br />
\]<br />
The first column of \(T\) means that if you cross a red with a pink, then \(50 \%\) of the time you'll get a red and \(50 \%\) a pink - and never get a white. The second column gives the result if you cross a pink with a pink (25\% red, \(50 \%\) pink, \(25 \%\) white) while the third column concerns crossing a white with a pink.<br />
<br />
In the long run, if you continue crossing the offsprings with only pink plants, what percentage of the three types of flowers would you expect to see in your garden?Mathematicshttps://mathsgee.com/36503/certain-species-dominant-recesive-flowers-depending-genotypeFri, 21 Jan 2022 10:06:42 +0000Multinational companies in the Americas, Asia, and Europe have assets of \(\$ 4\) trillion.
https://mathsgee.com/36502/multinational-companies-americas-europe-assets-trillion
Multinational companies in the Americas, Asia, and Europe have assets of \(\$ 4\) trillion. At the start, \(\$ 2\) trillion are in the Americas and \(\$ 2\) trillion are in Europe. Each year \(1 / 2\) of the Americas money stays home and \(1 / 4\) goes to each of Asia and Europe. For Asia and Europe, \(1 / 2\) stays home and \(1 / 2\) is sent to the Americas.<br />
1<br />
a) Let \(C_{k}\) be the column vector with the assets of the Americas, Asia, and Europe at the beginning of year \(k\). Find the transition matrix \(T\) that gives the amount in year \(k+1: C_{k+1}=T C_{k}\)<br />
b) Find the eigenvalues and eigenvectors of \(T\).<br />
c) Find the limiting distribution of the \(\$ 4\) trillion as the world ends<br />
d) Find the distribution of the \(\$ 4\) trillion at year \(k\).Mathematicshttps://mathsgee.com/36502/multinational-companies-americas-europe-assets-trillionFri, 21 Jan 2022 10:05:50 +0000Suppose there is an epidemic in which every month half of those who are well become sick, half of those who are sick get well, and a quarter of those who are sick die. Find the steady state for the corresponding Markov process
https://mathsgee.com/36501/suppose-epidemic-quarter-steady-corresponding-markov-process
Suppose there is an epidemic in which every month half of those who are well become sick, half of those who are sick get well, and a quarter of those who are sick die. Find the steady state for the corresponding Markov process<br />
\[<br />
\left(\begin{array}{l}<br />
w_{n+1} \\<br />
s_{n+1} \\<br />
d_{n+1}<br />
\end{array}\right)=\left(\begin{array}{ccc}<br />
\frac{1}{2} & \frac{1}{2} & 0 \\<br />
\frac{1}{2} & \frac{1}{4} & 0 \\<br />
0 & \frac{1}{4} & 1<br />
\end{array}\right)\left(\begin{array}{l}<br />
w_{n} \\<br />
s_{n} \\<br />
d_{n}<br />
\end{array}\right)<br />
\]Mathematicshttps://mathsgee.com/36501/suppose-epidemic-quarter-steady-corresponding-markov-processFri, 21 Jan 2022 10:03:55 +0000A long queue in front of a Moscow market in the Stalin era sees the butcher whisper to the first in line. He tells her "Yes, there is steak today." She tells the one behind her and so on down the line.
https://mathsgee.com/36500/front-moscow-market-stalin-butcher-whisper-first-tells-behind
A long queue in front of a Moscow market in the Stalin era sees the butcher whisper to the first in line. He tells her "Yes, there is steak today." She tells the one behind her and so on down the line. However, Moscow housewives are not reliable transmitters. If one is told "yes", there is only an \(80 \%\) chance she'll report "yes" to the person behind her. On the other hand, being optimistic, if one hears "no", she will report "yes" \(40 \%\) of the time.<br />
<br />
If the queue is very long, what fraction of them will hear "there is no steak"? [This problem can be solved without finding a formula for \(P^{n}\) (here \(P\) is the transition matrix) - although you might find it a challenge to find the formula].Mathematicshttps://mathsgee.com/36500/front-moscow-market-stalin-butcher-whisper-first-tells-behindFri, 21 Jan 2022 09:35:53 +0000Let \(T\) be the transition matrix of a Markov process. If \(P\) is a probability vector, show that \(T P\) is also a probability vector.
https://mathsgee.com/36499/transition-matrix-process-probability-probability-vector
Let \(T\) be the transition matrix of a Markov process. If \(P\) is a probability vector, show that \(T P\) is also a probability vector.Mathematicshttps://mathsgee.com/36499/transition-matrix-process-probability-probability-vectorFri, 21 Jan 2022 09:34:59 +0000Say you pick a point \((x, y)\) at random in the unit square \(0 \leq x \leq 1,0 \leq y \leq 1\). Compute the probability it will be in the set
https://mathsgee.com/36498/pick-point-random-unit-square-compute-probability-will-bein
Say you pick a point \((x, y)\) at random in the unit square \(0 \leq x \leq 1,0 \leq y \leq 1\). Compute the probability it will be in the set<br />
\[<br />
S=\{x \leq y, \quad \text { and } \quad x \leq 1 / 2, \quad \text { and } \quad y-x \leq 1 / 2, \quad \text { and } \quad 1-y \leq 1 / 2\}<br />
\]Mathematicshttps://mathsgee.com/36498/pick-point-random-unit-square-compute-probability-will-beinFri, 21 Jan 2022 09:34:14 +0000Say you pick a point \(x\) at random in the interval \(0 \leq x \leq 1\). What is the probbility that it will be in the set
https://mathsgee.com/36497/say-pick-point-random-interval-leq-what-probbility-that-will
Say you pick a point \(x\) at random in the interval \(0 \leq x \leq 1\). What is the probability that it will be in the set<br />
a) \(S=\{x \leq 1 / 2\) and \(x \geq 1 / 4\}\) ?<br />
b) \(T=\{x \leq 1 / 2\) and \(x \geq 3 / 4\} ?\)Mathematicshttps://mathsgee.com/36497/say-pick-point-random-interval-leq-what-probbility-that-willFri, 21 Jan 2022 09:33:26 +0000 A number, \(k\), of people are subjected to a blood test, the result of which is either "positive" or "negative". It can be processed in two ways:
https://mathsgee.com/36495/people-subjected-result-either-positive-negative-processed
A number, \(k\), of people are subjected to a blood test, the result of which is either "positive" or "negative". It can be processed in two ways:<br />
<br />
i). Each person can be tested separately, so \(k\) tests are required.<br />
ii). The blood samples of all \(k\) people can be pooled and analyzed together. If this test is negative, then one test suffices for the \(k\) people, while if the test is positive, each of the \(k\) people must be tested separately so \(k+1\) tests are then required.<br />
Assume that the probability, \(p\), that a test is positive is the same for all people and that these events are all independent.<br />
a). Find the probability that the test for a pooled sample of \(k\) people will be positive.<br />
b). What is the expected value of the number of tests necessary under plan ii)?Mathematicshttps://mathsgee.com/36495/people-subjected-result-either-positive-negative-processedFri, 21 Jan 2022 09:31:45 +0000Would you accept a gamble that offers a 10-percent chance to win \(\$ 95\) and a 90-percent chance to lose \$5?
https://mathsgee.com/36494/would-accept-gamble-offers-percent-chance-percent-chance-lose
a). Would you accept a gamble that offers a 10-percent chance to win \(\$ 95\) and a 90-percent chance to lose \$5?<br />
b). Would you pay \(\$ 5\) to participate in a lottery that offers a 10-percent chance to win \(\$ 100\) and a 90 -percent chance to win nothing?<br />
c). Do the options in a)-b) offer identical outcomes?<br />
Remarks:<br />
a). The great majority of people in the study rejected this proposition as a loser.<br />
b)-c). A large proportion of those who refused the first option accepted the second. But the options offer identical outcomes. As Kahneman and Tversky see it: "Thinking of the \(\$ 5\) as a payment makes the venture more acceptable than thinking of the same amount as a loss." It's all a matter of how the situation is framed in this case, the extent to which people are "risk averse."Mathematicshttps://mathsgee.com/36494/would-accept-gamble-offers-percent-chance-percent-chance-loseFri, 21 Jan 2022 09:30:21 +0000In an oft-cited experiment the psychologists Kahneman and Tversky asked a group of subjects to imagine the outbreak of an unusual disease, expected to kill 600 people, and to choose between two public health programs to combat it.
https://mathsgee.com/36493/experiment-psychologists-kahneman-subjects-expected-programs
In an oft-cited experiment the psychologists Kahneman and Tversky asked a group of subjects to imagine the outbreak of an unusual disease, expected to kill 600 people, and to choose between two public health programs to combat it.<br />
Program A, the subjects were told, had 400 people would die.<br />
Program B had a one-third probability that no one would die and a two-thirds probability that 600 people would die<br />
Which would you choose? Explain your decision.Mathematicshttps://mathsgee.com/36493/experiment-psychologists-kahneman-subjects-expected-programsFri, 21 Jan 2022 09:28:08 +0000[Monty Hall Problem] Suppose you're on a game show, and you're given the choice of three doors:
https://mathsgee.com/36492/monty-problem-suppose-youre-youre-given-choice-three-doors
[Monty Hall Problem] Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to switch?<br />
What should you do -- and why? A "tree diagram" may be useful.Mathematicshttps://mathsgee.com/36492/monty-problem-suppose-youre-youre-given-choice-three-doorsFri, 21 Jan 2022 09:26:40 +0000In the Cancer Test Case, we used that everyone who has the cancer tests positive. Instead, say \(5 \%\) of those who have the cancer test negative (false negative). Of those who test negative, what is the likelihood that they have this cancer?
https://mathsgee.com/36491/everyone-positive-negative-negative-negative-likelihood
In the Cancer Test Case, we used that everyone who has the cancer tests positive. Instead, say \(5 \%\) of those who have the cancer test negative (false negative). Of those who test negative, what is the likelihood that they have this cancer?Mathematicshttps://mathsgee.com/36491/everyone-positive-negative-negative-negative-likelihoodFri, 21 Jan 2022 09:25:16 +0000Answered: You are interviewing 6 candidates for a job. As you proceed, you determine the relative ranks of the candidates (you won't know the "true rank" until you have interviewed all of them).
https://mathsgee.com/36489/interviewing-candidates-determine-candidates-interviewed?show=36490#a36490
SOLUTION: Using this strategy you win if the second best candidate is in the first group of 5 and the best candidate is in the last group of 5, so \(25 \%\) of the time.Mathematicshttps://mathsgee.com/36489/interviewing-candidates-determine-candidates-interviewed?show=36490#a36490Fri, 21 Jan 2022 09:24:22 +0000In a certain town there are 2 bus companies whose buses stop at the Main Street station. One company's busses run every 8 minutes, the other runs every 10 minutes - but the times of arrival of the previous busses are unknown.
https://mathsgee.com/36488/companies-station-companys-minutes-minutes-previous-unknown
a) In a certain town there are 2 bus companies whose buses stop at the Main Street station. One company's busses run every 8 minutes, the other runs every 10 minutes - but the times of arrival of the previous busses are unknown. The question is, what is the average length of time that you will wait for a bus after at the station?<br />
b). Same as the above, but 3 bus companies - - whose busses each stop every 10 minutes.Mathematicshttps://mathsgee.com/36488/companies-station-companys-minutes-minutes-previous-unknownFri, 21 Jan 2022 09:23:08 +0000A big dart board consists of three concentric disks of radius 1,2, and 3 feet. If a dart lands in the center disk you get 50 points. If it lands in the middle ring you get 25 points, while if it lands in the outer ring you get 10 points.
https://mathsgee.com/36487/consists-concentric-radius-center-points-middle-points-points
<p>A big dart board consists of three concentric disks of radius 1,2, and 3 feet. If a dart lands in the center disk you get 50 points. If it lands in the middle ring you get 25 points, while if it lands in the outer ring you get 10 points.
<br>
Compute both the expected value and standard deviation of the number of points you'l1 get by throwing darts at random at this board.</p>
<p><img alt="target" src="https://mathsgee.com/?qa=blob&qa_blobid=11221528235443449352" style="height:200px; width:233px"></p>
<p> </p>Mathematicshttps://mathsgee.com/36487/consists-concentric-radius-center-points-middle-points-pointsFri, 21 Jan 2022 09:22:10 +0000The next three players in a game win \(30 \%, 20 \%\) and \(25 \%\) of the time, respectively. What is the likelihood that none of them will win this time?
https://mathsgee.com/36486/three-players-game-time-respectively-likelihood-this-time
The next three players in a game win \(30 \%, 20 \%\) and \(25 \%\) of the time, respectively. What is the likelihood that none of them will win this time?<br />
[Equivalent wording: It is the fifth inning of a baseball game. The batting averages of the next three batters are \(.300 . .200\), and \(.250\). Say they face an average pitcher. What is the likelihood that none of them will get a hit this inning?]Mathematicshttps://mathsgee.com/36486/three-players-game-time-respectively-likelihood-this-timeFri, 21 Jan 2022 09:20:34 +0000A well-known folk legend. On the night before the big chemistry exam, two students were partying all night -- and over slept the exam.
https://mathsgee.com/36485/known-legend-before-chemistry-students-partying-night-slept
A well-known folk legend. On the night before the big chemistry exam, two students were partying all night -- and over slept the exam. Their excuse to the professor was that they were driving from some friends who lived far away and had a flat tire; could they please take a make-up exam. The professor agreed. She wrote an exam and put them in different rooms to take it. The first question, worth 5 points, was easy. The second question, on the other side of the page was worth 95 points. It simply asked, "Which tire was it?".<br />
What is the probability that both students would give the same answer?Mathematicshttps://mathsgee.com/36485/known-legend-before-chemistry-students-partying-night-sleptFri, 21 Jan 2022 09:19:45 +0000In an office, say your personal code to use the photo copy machine is the last four digits of your social security number.
https://mathsgee.com/36484/office-personal-photo-machine-digits-social-security-number
In an office, say your personal code to use the photo copy machine is the last four digits of your social security number.<br />
<br />
a). If 100 people share that copier, what is the probability that at least two people have the same code?<br />
<br />
b). Same question with 150 people.Mathematicshttps://mathsgee.com/36484/office-personal-photo-machine-digits-social-security-numberFri, 21 Jan 2022 09:18:50 +0000Say you toss a pair of dice \(N\) times. You win if a pair of 1 's appear on the same toss. What is the smallest value of \(N\) so your likelihood of winning is more than \(50 \%\) ?
https://mathsgee.com/36483/pair-dice-times-appear-smallest-value-likelihood-winning-more
Say you toss a pair of dice \(N\) times. You win if a pair of 1 's appear on the same toss. What is the smallest value of \(N\) so your likelihood of winning is more than \(50 \%\) ?Mathematicshttps://mathsgee.com/36483/pair-dice-times-appear-smallest-value-likelihood-winning-moreFri, 21 Jan 2022 09:17:46 +0000Genetically, to whom are you more closely related:
https://mathsgee.com/36482/genetically-to-whom-are-you-more-closely-related
Genetically, to whom are you more closely related:<br />
a). Your brother or your mother?<br />
b). Your granddaughter or your sister's son?<br />
c). Your aunt or your brother's daughter?<br />
d). Your uncle or your first cousin's son?<br />
e). In a court case, say there the deceased has no surviving children, grand children, ... (direct lineal descendants) and no will. But there are aunts, uncles, and cousins. The estate has lots of money. What algorithm might the Court use to distribute the legacy?Mathematicshttps://mathsgee.com/36482/genetically-to-whom-are-you-more-closely-relatedFri, 21 Jan 2022 09:16:57 +0000Given a date, such as March 17,2011 , determine the day of the week (Monday, Tuesday, etc.). [Hint: If today is a Saturday, a day 700 days in the future is also a Saturday, while a day 702 days in the future is "clearly" a Monday (why?)].
https://mathsgee.com/36481/determine-monday-tuesday-saturday-saturday-future-clearly
Given a date, such as March 17,2011 , determine the day of the week (Monday, Tuesday, etc.). [Hint: If today is a Saturday, a day 700 days in the future is also a Saturday, while a day 702 days in the future is "clearly" a Monday (why?)].Mathematicshttps://mathsgee.com/36481/determine-monday-tuesday-saturday-saturday-future-clearlyFri, 21 Jan 2022 09:15:46 +0000Answered: Consider a stock that is sold on the New York Stock Exchange. If someone is Paris places a buy order and \(1 / 100\) of a second later someone in New York buys the same stock, which order is executed first?
https://mathsgee.com/36479/consider-exchange-someone-places-second-someone-executed?show=36480#a36480
\(\begin{aligned} \text { Speed of light in a vacuum } & \approx 300,000 \mathrm{~km} / \mathrm{sec} \\ & \approx 186,000 \mathrm{miles} / \mathrm{sec} \\ & \approx 1 \text { foot } / \mathrm{ns} \approx 1 \mathrm{~m} / 3.3 \mathrm{~ns} \quad(\mathrm{~ns}=\text { nanoseconds }) \\ & \approx 4,000 \mathrm{miles} / .0215 \mathrm{sec} \quad(3,625 \text { miles from Paris to New York }\end{aligned}\)Mathematicshttps://mathsgee.com/36479/consider-exchange-someone-places-second-someone-executed?show=36480#a36480Fri, 21 Jan 2022 09:14:06 +0000Answered: Let \(S_{N}:=1+\frac{1}{2}+\cdots+\frac{1}{N}\). Find an estimate for \(N\) so that \(S_{N}>100\).
https://mathsgee.com/36477/let-s-1-frac-1-2-cdots-frac-1-find-an-estimate-for-so-that-s-100?show=36478#a36478
By the idea behind the integral test<br />
\[<br />
\ln (N+1)<S_{N}<1+\ln N<br />
\]<br />
Thus, to insure that \(S_{N}>100\) pick \(\ln (N+1)>100\), that is, \(N+1>\) \(e^{100} \approx 2.7 * 10^{43}\).<br />
On the other hand, if \(1+\ln N<100\), then \(S_{N}<100\). Here we can pick any \(N<e^{99} \approx 10^{43}\).Mathematicshttps://mathsgee.com/36477/let-s-1-frac-1-2-cdots-frac-1-find-an-estimate-for-so-that-s-100?show=36478#a36478Fri, 21 Jan 2022 09:12:13 +0000Answered: List these numbers from smallest to largest: \(2^{121}, 9^{55}, 7^{88}\)
https://mathsgee.com/36473/list-these-numbers-from-smallest-to-largest-2-121-9-55-7-88?show=36476#a36476
- \(2^{121}=\left(2^{11}\right)^{11}=2048^{11} \approx 2.710^{36}\)<br />
- \(9^{55}=\left(9^{5}\right)^{11}=32805^{11} \approx 3.010^{52}\)<br />
- \(7^{88}=\left(7^{8}\right)^{11}=5764801^{11} \approx 2.310^{74}\)Mathematicshttps://mathsgee.com/36473/list-these-numbers-from-smallest-to-largest-2-121-9-55-7-88?show=36476#a36476Fri, 21 Jan 2022 09:10:54 +0000Answered: What is the age of the universe (14 billion years) in seconds?
https://mathsgee.com/36474/what-is-the-age-of-the-universe-14-billion-years-in-seconds?show=36475#a36475
\(\mathrm{N}:=\) number of seconds since the birth of our universe.<br />
<br />
Age of the universe (14 billion years) in seconds:<br />
\[<br />
\begin{aligned}<br />
N &=14 * 10^{9} \text { years } * \frac{365 \text { days }}{\text { year }} * \frac{24 \text { hours }}{\text { day }} * \frac{3600 \text { seconds }}{\text { hour }} \\<br />
&=441,504,000 * 10^{9} \approx 4.4 * 10^{17} \text { seconds. }<br />
\end{aligned}<br />
\]Mathematicshttps://mathsgee.com/36474/what-is-the-age-of-the-universe-14-billion-years-in-seconds?show=36475#a36475Fri, 21 Jan 2022 09:10:08 +0000A tridiagonal matrix is a square matrix with zeroes everywhere except on the main diagonal and the diagonals just above and below the main diagonal.
https://mathsgee.com/36472/tridiagonal-square-everywhere-diagonal-diagonals-diagonal
A tridiagonal matrix is a square matrix with zeroes everywhere except on the main diagonal and the diagonals just above and below the main diagonal.<br />
<br />
Let \(T\) be a real anti-symmetric tridiagonal matrix with elements \(t_{12}=c_{1}, t_{23}=c_{2}, \ldots\), \(t_{n-1 n}=c_{n-1}\). If \(n\) is even, compute det \(T\).Mathematicshttps://mathsgee.com/36472/tridiagonal-square-everywhere-diagonal-diagonals-diagonalFri, 21 Jan 2022 08:48:50 +0000One year on July 1 in northern Canada I was camping and the sun set around 11 PM. A year later in Hawaii the sun set at around 7 PM.
https://mathsgee.com/36471/year-july-northern-canada-camping-around-later-hawaii-around
One year on July 1 in northern Canada I was camping and the sun set around 11 PM. A year later in Hawaii the sun set at around 7 PM. This led to the basic question, given the day of the year and the latitude, when does the sun set? [Since this depends on "time zones", a better question is probably, "how many hours of daylight are there?"Mathematicshttps://mathsgee.com/36471/year-july-northern-canada-camping-around-later-hawaii-aroundFri, 21 Jan 2022 08:48:03 +0000Let \(v_{1} \ldots v_{k}\) be vectors in a linear space with an inner product \(\langle,\),\(rangle . Define the\) Gram determinant by \(G\left(v_{1}, \ldots, v_{k}\right)=\operatorname{det}\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).
https://mathsgee.com/36470/vectors-product-define-determinant-operatorname-langle-rangle
Let \(v_{1} \ldots v_{k}\) be vectors in a linear space with an inner product \(\langle,\),\(rangle . Define the\) Gram determinant by \(G\left(v_{1}, \ldots, v_{k}\right)=\operatorname{det}\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).<br />
a) If the \(v_{1} \ldots v_{k}\) are orthogonal, compute their Gram determinant.<br />
b) Show that the \(v_{1} \ldots v_{k}\) are linearly independent if and only if their Gram determinant is not zero.<br />
c) Better yet, if the \(v_{1} \ldots v_{k}\) are linearly independent, show that the symmetric matrix \(\left(\left\langle v_{i}, v_{j}\right\rangle\right)\) is positive definite. In particular, the inequality \(G\left(v_{1}, v_{2}\right) \geq 0\) is the Schwarz inequality.<br />
d) Conversely, if \(A\) is any \(n \times n\) positive definite matrix, show that there are vectors \(v_{1}, \ldots, v_{n}\) so that \(A=\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).<br />
e) Let \(\mathcal{S}\) denote the subspace spanned by the linearly independent vectors \(w_{1} \ldots w_{k} .\) If \(X\) is any vector, let \(P_{\mathcal{S}} X\) be the orthogonal projection of \(X\) into \(\mathcal{S}\), prove that the distance \(\left\|X-P_{\mathcal{S}} X\right\|\) from \(X\) to \(\mathcal{S}\) is given by the formula<br />
\[<br />
\left\|X-Z_{\mathcal{S}} X\right\|^{2}=\frac{G\left(X, w_{1}, \ldots, w_{k}\right)}{G\left(w_{1}, \ldots, w_{k}\right)} .<br />
\]Mathematicshttps://mathsgee.com/36470/vectors-product-define-determinant-operatorname-langle-rangleFri, 21 Jan 2022 08:46:45 +0000Let \(Z_{1}, \ldots, Z_{k}\) be distinct points in \(\mathbb{R}^{n}\). Find a unique point \(X_{0}\) in \(\mathbb{R}^{n}\) at which the function \[ Q(X)=\left\|X-Z_{1}\right\|^{2}+\cdots+\left\|X-Z_{k}\right\|^{2} \]
https://mathsgee.com/36469/ldots-distinct-points-mathbb-unique-mathbb-function-right
Let \(Z_{1}, \ldots, Z_{k}\) be distinct points in \(\mathbb{R}^{n}\). Find a unique point \(X_{0}\) in \(\mathbb{R}^{n}\) at which the function<br />
\[<br />
Q(X)=\left\|X-Z_{1}\right\|^{2}+\cdots+\left\|X-Z_{k}\right\|^{2}<br />
\]<br />
achieves its minimum value by "completing the square" to obtain the identity<br />
\[<br />
Q(X)=k\left\|X-\frac{1}{k} \sum_{n=1}^{k} Z_{n}\right\|^{2}+\sum_{j=1}^{k}\left\|\left.Z_{j}\right|^{2}-\frac{1}{k}\right\| \sum_{j=1}^{k} Z_{j} \|^{2}<br />
\]Mathematicshttps://mathsgee.com/36469/ldots-distinct-points-mathbb-unique-mathbb-function-rightFri, 21 Jan 2022 08:46:01 +0000Answered: Let \(A\) be a positive definite \(n \times n\) real matrix, \(\vec{b}\) a real vector, and \(\vec{N}\) a real unit vector.
https://mathsgee.com/36467/positive-definite-times-matrix-real-vector-real-unit-vector?show=36468#a36468
(a) \(c \leq\left\langle b, A^{-1} b\right\rangle\). If \(n=1\), this of course reduces to a familiar condition.<br />
<br />
(b) \[<br />
d=-\left\langle\vec{N}, A^{-1} \vec{b}\right\rangle \pm \sqrt{\left\langle\vec{N}, A^{-1} \vec{N}\right\rangle} \sqrt{\left\langle\vec{b}, A^{-1} \vec{b}\right\rangle-c} .<br />
\]<br />
For \(n=1\) this is just the solution \(d=\frac{-b \pm \sqrt{b^{2}-a c}}{a}\) of the quadratic equation \(\left.a x^{2}+2 b x+c=0 .\right]\)Mathematicshttps://mathsgee.com/36467/positive-definite-times-matrix-real-vector-real-unit-vector?show=36468#a36468Fri, 21 Jan 2022 08:44:11 +0000Let \(\vec{x}\) and \(\vec{p}\) be points in \(\mathbb{R}^{n}\). Under what conditions on the scalar \(c\) is the set \[ \|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0 \]
https://mathsgee.com/36466/points-mathbb-under-what-conditions-the-scalar-langle-rangle
a) Let \(\vec{x}\) and \(\vec{p}\) be points in \(\mathbb{R}^{n}\). Under what conditions on the scalar \(c\) is the set<br />
\[<br />
\|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0<br />
\]<br />
a sphere \(\left\|\vec{x}-\vec{x}_{0}\right\|=R \geq 0\) ? Compute the center, \(\vec{x}_{0}\), and radius, \(R\), in terms of \(\vec{p}\) and \(c\).<br />
b) Let<br />
\[<br />
\begin{aligned}<br />
Q(\vec{x}) &=\sum a_{i j} x_{i} x_{j}+2 \sum b_{i} x_{i}+c \\<br />
&=\langle\vec{x}, A \vec{x}\rangle+2\langle\vec{b}, \vec{x}\rangle+c<br />
\end{aligned}<br />
\]<br />
be a real quadratic polynomial so \(\vec{x}=\left(x_{1}, \ldots, x_{n}\right), \vec{b}=\left(b_{1}, \ldots, b_{n}\right)\) are real vectors and \(A=\left(a_{i j}\right)\) is a real symmetric \(n \times n\) matrix. Just as in the case \(n=1\) (which you should do first), if \(A\) is invertible find a vector \(\vec{v}\) (depending on \(A\) and \(\vec{b}\) ) so that the change of variables \(\vec{y}==\vec{x}-\vec{v}\) (this is a translation by the vector \(\vec{v}\) ) so that in the new \(\vec{y}\) variables \(Q\) has the simpler form<br />
\[<br />
Q=\langle\vec{y}, A \vec{y}\rangle+\gamma \text { that is, } Q=\sum a_{i j} y_{i} y_{j}+\gamma,<br />
\]<br />
where \(\gamma=c-\left\langle\vec{b}, A^{-1} \vec{b}\right\rangle\).<br />
As an example, apply this to \(Q(\vec{x})=2 x_{1}^{2}+2 x_{1} x_{2}+3 x_{2}-4\).Mathematicshttps://mathsgee.com/36466/points-mathbb-under-what-conditions-the-scalar-langle-rangleFri, 21 Jan 2022 08:39:47 +0000Let \(V \subset \mathbb{R}^{n}\) be a linear space, \(Q: R^{n} \rightarrow V^{\perp}\) the orthogonal projection into \(V^{\perp}\), and \(x \in \mathbb{R}^{n}\) a given vector.
https://mathsgee.com/36465/subset-mathbb-rightarrow-orthogonal-projection-mathbb-vector
[Dual variational problems] Let \(V \subset \mathbb{R}^{n}\) be a linear space, \(Q: R^{n} \rightarrow V^{\perp}\) the orthogonal projection into \(V^{\perp}\), and \(x \in \mathbb{R}^{n}\) a given vector. Note that \(Q=I-P\), where \(P\) in the orthogonal projection into \(V\)<br />
a) Show that \(\max _{\{z \perp V,\|z\|=1\}}\langle x, z\rangle=\|Q x\|\).<br />
b) Show that \(\min _{v \in V}\|x-v\|=\|Q x\|\).<br />
[Remark: dual variational problems are a pair of maximum and minimum problems whose extremal values are equal.]Mathematicshttps://mathsgee.com/36465/subset-mathbb-rightarrow-orthogonal-projection-mathbb-vectorFri, 21 Jan 2022 08:38:58 +0000Let \(V \subset \mathbb{R}^{n}\) be a subspace and \(Z \in \mathbb{R}^{n}\) a given vector. Find a unit vector \(X\) that is perpendicular to \(V\) with \(\langle X, Z\rangle\) as large as possible.
https://mathsgee.com/36464/subset-mathbb-subspace-perpendicular-langle-rangle-possible
a) Let \(V \subset \mathbb{R}^{n}\) be a subspace and \(Z \in \mathbb{R}^{n}\) a given vector. Find a unit vector \(X\) that is perpendicular to \(V\) with \(\langle X, Z\rangle\) as large as possible.<br />
b) Compute max \(\int_{-1}^{1} x^{3} h(x) d x\) where \(h(x)\) is any continuous function on the interval \(-1 \leq x \leq 1\) subject to the restrictions<br />
\[<br />
\int_{-1}^{1} h(x) d x=\int_{-1}^{1} x h(x) d x=\int_{-1}^{1} x^{2} h(x) d x=0 ; \quad \int_{-1}^{1}|h(x)|^{2} d x=1 .<br />
\]<br />
36<br />
c) Compute \(\min _{a, b, c} \int_{-1}^{1}\left|x^{3}-a-b x-c x^{2}\right|^{2} d x\).Mathematicshttps://mathsgee.com/36464/subset-mathbb-subspace-perpendicular-langle-rangle-possibleFri, 21 Jan 2022 08:37:00 +0000Find the function \(f \in \operatorname{span}\{1 \sin x, \cos x\}\) that minimizes \(\|\sin 2 x-f(x)\|\), where the norm comes from the inner product
https://mathsgee.com/36463/function-operatorname-minimizes-where-comes-inner-product
Find the function \(f \in \operatorname{span}\{1 \sin x, \cos x\}\) that minimizes \(\|\sin 2 x-f(x)\|\), where the norm comes from the inner product<br />
\[<br />
\langle f, g\rangle:=\int_{-\pi}^{\pi} f(x) g(x) d x \quad \text { on } \quad C[-\pi, \pi] .<br />
\]Mathematicshttps://mathsgee.com/36463/function-operatorname-minimizes-where-comes-inner-productFri, 21 Jan 2022 08:35:56 +0000Let \(C[-1,1]\) be the real inner product space consisting of all continuous functions \(f:[-1,1] \rightarrow \mathbb{R}\), with the inner product \(\langle f, g\rangle:=\int_{-1}^{1} f(x) g(x) d x\).
https://mathsgee.com/36462/product-consisting-continuous-functions-rightarrow-product
Let \(C[-1,1]\) be the real inner product space consisting of all continuous functions \(f:[-1,1] \rightarrow \mathbb{R}\), with the inner product \(\langle f, g\rangle:=\int_{-1}^{1} f(x) g(x) d x\). Let \(W\) be the subspace of odd functions, i.e. functions satisfying \(f(-x)=-f(x)\). Find (with proof) the orthogonal complement of \(W\).Mathematicshttps://mathsgee.com/36462/product-consisting-continuous-functions-rightarrow-productFri, 21 Jan 2022 08:34:34 +0000Let \(\mathcal{P}_{2}\) be the space of polynomials \(p(x)=a+b x+c x^{2}\) of degree at most 2 with the inner product \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\).
https://mathsgee.com/36461/mathcal-space-polynomials-degree-inner-product-langle-rangle
Let \(\mathcal{P}_{2}\) be the space of polynomials \(p(x)=a+b x+c x^{2}\) of degree at most 2 with the inner product \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\). Let \(\ell\) be the functional \(\ell(p):=p(0)\). Find \(h \in \mathcal{P}_{2}\) so that \(\ell(p)=\langle h, p\rangle\) for all \(p \in \mathcal{P}_{2}\).Mathematicshttps://mathsgee.com/36461/mathcal-space-polynomials-degree-inner-product-langle-rangleFri, 21 Jan 2022 08:33:56 +0000Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.
https://mathsgee.com/36460/let-mathcal-p-2-be-the-space-of-quadratic-polynomials
Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.<br />
<br />
a) Show that \(\langle f, g\rangle=f(-1) g(-1)+f(0) g(0)+f(1) g(1)\) is an inner product for this space.<br />
<br />
b) Using this inner product, find an orthonormal basis for \(\mathcal{P}_{2}\).<br />
<br />
c) Is this also an inner product for the space \(\mathcal{P}_{3}\) of polynomials of degree at most three? Why?Mathematicshttps://mathsgee.com/36460/let-mathcal-p-2-be-the-space-of-quadratic-polynomialsFri, 21 Jan 2022 08:33:18 +0000Answered: Using the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\), for which values of the real constants \(\alpha, \beta, \gamma\) are the quadratic polynomials ...
https://mathsgee.com/36457/product-langle-rangle-values-constants-quadratic-polynomials?show=36459#a36459
ANSWER:<br />
<br />
\( p_{2}(x)=x, p_{3}(x)=x^{2}-1 / 3 \)Mathematicshttps://mathsgee.com/36457/product-langle-rangle-values-constants-quadratic-polynomials?show=36459#a36459Fri, 21 Jan 2022 08:31:50 +0000Using the inner product of the previous problem, let \(\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}\) be an orthogonal basis for the space \(\mathcal{P}_{2}\) of quadratic polynomials and . . .
https://mathsgee.com/36458/product-previous-problem-orthogonal-quadratic-polynomials
Using the inner product of the previous problem, let \(\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}\) be an orthogonal basis for the space \(\mathcal{P}_{2}\) of quadratic polynomials and let \(\mathcal{S}=\operatorname{span}\left(x, x^{2}\right) \subset\) \(\mathcal{P}_{2}\). Using the basis \(\mathcal{B}\), find the linear map \(P: \mathcal{P}_{2} \rightarrow \mathcal{P}_{2}\) that is the orthogonal projection from \(\mathcal{P}_{2}\) onto \(\mathcal{S}\).Mathematicshttps://mathsgee.com/36458/product-previous-problem-orthogonal-quadratic-polynomialsFri, 21 Jan 2022 08:30:42 +0000In a complex vector space (with a hermitian inner product), if a matrix \(A\) satisfies \(\langle X, A X\rangle=0\) for all vectors \(X\), show that \(A=0\). [The previous problem shows that this is false in a real vector space].
https://mathsgee.com/36456/complex-hermitian-product-satisfies-vectors-previous-problem
In a complex vector space (with a hermitian inner product), if a matrix \(A\) satisfies \(\langle X, A X\rangle=0\) for all vectors \(X\), show that \(A=0\). [The previous problem shows that this is false in a real vector space].Mathematicshttps://mathsgee.com/36456/complex-hermitian-product-satisfies-vectors-previous-problemFri, 21 Jan 2022 08:27:16 +0000Let \(L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be a linear map with the property that \(L \mathbf{v} \perp \mathbf{v}\) for every \(\mathbf{v} \in \mathbb{R}^{3}\).
https://mathsgee.com/36455/mathbb-rightarrow-mathbb-linear-property-mathbf-mathbf-mathbb
Let \(L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be a linear map with the property that \(L \mathbf{v} \perp \mathbf{v}\) for every \(\mathbf{v} \in \mathbb{R}^{3}\). Prove that \(L\) cannot be invertible.<br />
Is a similar assertion true for a linear map \(L: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) ?Mathematicshttps://mathsgee.com/36455/mathbb-rightarrow-mathbb-linear-property-mathbf-mathbf-mathbbFri, 21 Jan 2022 08:26:31 +0000