Let P3 have the inner product as in Exercise 9, with p0, p1, and q the polynomials described there. Find the best approximation to by polynomials in Span .
Let P3 have the inner product given by evaluation at
a. Compute the orthogonal projection of p2 onto the subspace spanned by p0 and p1.
b. Find a polynomial q that is orthogonal to p0 and p1, such that is an orthogonal basis for Span Scale the polynomial q so that its vector of values at .
Violence, Warfare, Destruction Friday, March 18, 2016 8:38 PM The Trojan Saga and the Illiad The Illiad is not necessarily about the Trojan war Poem that is about Achilles and his rage Back story of the Trojan War Zeus visited Leda in the form of a swan and they had two children from one eggs One egg held Polydeuces and Helen - Zeus The other held Castor and Clytemnestra - Tyndareus Another version is that Helen is the daughter of Zeus and Nemesis (means someone who thinks they are better than the gods) Devine retribution She is born as a punishment for people who do not respect the gods Kings that come to Sparta for Helen's hand in marriage Menelaus, Diomedes, Patroclus, Odysseus (he takes himself out of the running when seeing the competition) They try to kill each other Attraction can cause violence Every pursuer of Helen must defend the winning of Helen The king choses Menelaus The Dioscuri - sons of Zeus They were honored first in Sparta then spread to Rome Saviors of people on earth and swift moving ships Caster and Polydeuces are not part of the sage of the trojan war Caster is known as a tamer of horses - mortal Polydeuces is a killed boxer - immortal They quarreled with the two sons of Aphareus (Idas and Lynceus) over the division of cattle Lynceus and Castor were killed Idas was destroyed by Zeus's thunderbolt Plydeuces prayed to Zeus that he would die with Castor Zeus gave him a choice and he choose that they would both live but spend alternate days on Olympus and Hades Helen Married Menelaus and had a daughter named Hermione Paris of Troy seduced her and took her back to Troy with him Achaean (Mycenaean Greek) expedition led by Agamemnon raised against Troy The judgement of Paris First version, the gods were guest at Peleus and Thetis's wedding Eris, the goddess of discord threw the golden apple on the table and said for the most beautiful Paris had to settle who was prettier between Athena, Hera, and Aphrodite He led them to Mt Ida each offered a gift and he choose Aphrodite who offered Helen This is how the war began Another version Hermes takes the apple and the three goddess to Paris who is tending his flock on Mt Ida He examines the goddesses and makes them get naked Aphrodite wins by promising to use love and desire to make Helen love him and come back to Troy with him Menelaus is mad because Paris has taken away his dignity Agamemnon and Menelaus are made at Paris not Helen All of the losers of the contest for Helen must defend Troy Other heros put into battle Odysseus - most clever hero but he did not want to fight the war An oracle tells him that if he goes he wont come back for 20 years He plows his own land and puts salt into it Palamedes - goes to check if he is insane Puts Telemachus (Odysseus's son) in front of his horses Odysseus saves his son so is forced to fight Achilles Terrifying warrior, Greek gentleman Well educated, musician, eloquent speaker - Chiron Greeks viewed war as one side of human existence The other side is being an educated gentleman He was hidden and disguised as a girl to avoid the war on Scyros Odysseus brought gifts and when Achilles liked the spear and shield he was discovered Now Achilles has to go on the expedition to Troy Achilles has a son while on Scyros Achilles is only about 10 Sacrifice Aulis kills a sacred stag and Artemis is mad To solve this he must sacrifice his daughter Iphigeneia Orders her to be brought to him and kills her Then the Greeks sail to Troy Alternate version - a magic stag flies down last minute and rescues her Takeaways Desire and violence are linked Desire brings retribution Men are willing to commit violence to receive sexual retribution Helen is a personification of Desire Honor and Violence Destiny and violence Achilles is destined for violence The apple came from the goddess of strife which led to the Trojan war Trojan War Achilles vs Agamemnon Nine years of war There had a lot of water and supplies so Troy survived for 9 years The Greeks survived by plundering the villages around Troy Steal food, animals, women - sex slaves Achilles is young so these activities are all that he knows Agamemnon's prize "Chryseis" Her father Chryses begs for his daughter back He prays to Apollo to punish the Greeks for Agamemnon Apollo kills the Greek's cattle with his arrows The army begins to die too - retribution for keeping Chryseis The Greeks want Agamemnon to give her back Agamemnon agreed to giver her back but he took Achilles prize "Briseis" Achilles is ready to kill Agamemnon but Athena stops him from killing her Achilles decides to leave to punish Agamemnon Achilles cries about Briseis being taken away from him Public crying is not seen as unmanly Thetis sees him and vows to punish Agamemnon She goes to Zeus and asks Zeus to punish the Greeks for dishonoring Achilles Zeus owes her because she got him out of Hera, Apollo and Hades trap Result: Hector and his troops become triumphant Hector is a family man - he loves his wife and child Hector pushes the Greeks back and sets fire to his ships Agamemnon realizes they are losing because of the loss of Achilles Odysseus begs Achilles to return to the war Achilles is playing the lyre when Odysseus and two other men arrive He stops and goes and makes them dinner Odysseus tells Achilles that Agamemnon will give him 10 women and 10 cities plus the plunder from the continued war Achilles does not fall for this - he doesn’t think its fair that he does all the fighting when Agamemnon gets all the glory and prizes Achilles goes home Patroclus' (Achilles's best friend) stays to fight - Achilles gives him his armor Patroclus goes on a rampage and makes it to the walls of Troy Because he was not destined to capture Troy so Apollo appears at the walls and takes off his armor Hector kills him and steals the armor Achilles is enraged and wants Hector to pay for Patroclus's death Thetis reminds him that he will die right after Hector does Thetis goes to Hephaestus and asks for new armor for Achilles Agamemnon blames what he did to Achilles on Zeus Achilles doesn’t care he just wants to avenge Patroclus Achilles goes on a very violet rampage Aeneas is the one person who escapes Achilles because is destined to be the founder of Rome Achilles is in such a rampage he takes on the river monster Hera sends Hephaestus to the river and evaporates it with fire to allow Achilles to get to Troy Hector vs Achilles Hector knows that if he goes and fights Achilles he will die but he would rather die than loss his army Hector runs away from Achilles and Achilles chases him around Troy Athena transforms into one of Hector's brothers and tells Hector that they will fight together Achilles throws his spear at Hector and misses, Athena brings it back to him Hector throws his spear and misses now he only has a sword They run at each other Achilles knows the weak spots on the armor because it is his own Achilles rams his spear into his throat - by killing Hector he is in a way killing himself Hector declares that though Achilles has won this battle, Paris will kill Achilles Achilles drags his body behind his chariot All of the woman of Troy wail about the death Funeral games for Patroclus Achilles sacrificed 12 Trojan youths - not normal part of the games Every day for 10 days Achilles drags Hectors body around the camp Apollo restores Hectors body every night Zeus sends Hermes to the Hector's father (Priam) and they go to the Greek camp and Priam asks Achilles for Hector's body - gets down on his knees kisses his hand and gives a speech comparing himself to Achilles father Achilles gives Priam his word that no one harm him as he celebrates Hector's funeral Three people give speeches His wife - Andromache, his mother - Hecuba, the woman who made it all happen - Helen Takeaways Is war glorious Warriors receive glory Necessary Does it bring destruction Mothers and children are left alone Parents lose their children Fall of Troy The amazons come to help the Trojans defend themselves against the Greeks The amazon queen Penthesilea kills Achilles but Thetis revives him Achilles kills Penthesilea and falls in love as his is killing her Again showing the anguish with war - tragedy The Trojans also call us the Ethiopians Paris is archer - shows that he is a coward Apollo guides Paris's arrow into Achilles heel When Achilles is dead Ajax fights off the Trojans who want the armor and Odysseus takes the armor to safety Agamemnon sends spies to listen to the Trojan warriors to see who they fear the most Odysseus wins the argument of who should get it because he is more fearful Ajax goes crazy and hacks up sheep that he thinks are Agamemnon and Odysseus He is so ashamed that he went crazy He sets up a sword and falls on it to kill himself Violence can make people see red, and lose all sanity and even kill themselves Odysseus captures a Trojan profit which tells him that the Greeks need two more heroes to win the war Philoctetes - Odysseus goes to Lemnos and persuades him to come to war He is a great archer He kills Paris Neoptolemus - Achilles's son The Trojan Horse Huge hollow wooden horse which holds the best warriors The Greeks leave the horse and sail out of sight Except for Sinon - he says that they left him and the horse is a peace offering Cassandra tells everyone that Greeks were going to come out of the horse but no one believes her Snakes choke and sink into the ground a priest and his sons who try to say that they shouldn’t take the horse Helen pretends to be the warriors wives but Odysseus tells the warriors it is a trick The Sack of Troy When the Trojans are asleep Sinon opens the trap door The Greeks open the wall of Troy The Greeks kill the Trojans Neoptolemus kills Priam like a sacrifice Aenaus escapes from the slaughter With his father Anchises Cassandra tries to take refuge in the temple of Apollo A different Ajax takes her out of the temple and rapes her Hecuba becomes a slave woman for one of the warriors Polyxena - daughter of Hecuba and Priam She is sacrificed to Achilles because he has been interested in her Menolaus searches for Helen - he wants to kill her but is so in love that he just takes her back to Sparta Takeaways War is glorious the victors get honor and wealth War is destructive the defeated are devastated War is tragic! Many great heroes die or are harmed Avoidable consequences Decisions are made even though the outcome is known Ajax does not have to kill himself over Achilles armor Achilles does not have to avenge Patrolus The Greeks do not have to fight so hard against the Trojans War is necessary! The gods bring it about Zeus started it all by making Paris make the decision about the beautiful goddess Apollo makes sure that Achilles dies Athena makes sure that Hector dies Gods will never die but they help continuously to make this war happen The consequences of war continue as the Greeks return home Irish Hero - Cuchulainn Friday, March 25, 2016 11:05 AM Believed in myth for a long period of time Decentralized country Never conquered by Rome Hunting, beer drinking, sex Druids religion Human sacrifice Believed there were invasions by gods who mated and gave birth to different heroes Cuchulainn Miraculous birth Mother is Cathbad -Daughter of a Druid Lugh flew into Cathbad's drink as a fly and impregnated her Childhood exploits Chuchulainn at the age of four knocks out 50 boys and defends himself from clubs and knives Culann's dog attacked him and he wrestled the dog off of him He had a reputation of a violent warrior at a young age When Cuchulainn heard the prophesy that whoever took up arms who live a short life but a heroic one Chuchulainn decides to go into battle at age 7 In battle Cuchulainn gets into a battle rage - like Achilles or Odin (Scadenavian) His muscles swell, he howls, one eye bulges He had to be doused in cool water to calm himself He has to turn away from woman to be doused in the water He collected the heads of warriors that he killed After his youth Cuchulainn fights a number of wars Connaught - queen Medb (shape shifter) Medb sends the people to capture a large cow Cuchulainn wins in battle and spares Medb's life but she has to return the bull The bull dies soon after Cuchulainn goes into battle because three kings are burning his crops and killing his cattle Cuchulainn throws spears but the kings throw them back and kills him He ties himself to a stone or tree to make so that he remains standing as he dies Takeaways Some humans are violent and destructive by nature He is a warrior from birth He is divine so this can add to his violence Constant war brings a tragic, premature death even to the best warrior Are the wars he fights necessary It was a defensive war Medb's men were destroying crops and cattle Deoes his patriotic motivation matter in judging his decisions to fights