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VectorCalculus&LinearAlgebra I

by: Maye Funk

VectorCalculus&LinearAlgebra I MATH 230

Marketplace > Yale University > Mathematics (M) > MATH 230 > VectorCalculus LinearAlgebra I
Maye Funk
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This 11 page Class Notes was uploaded by Maye Funk on Thursday October 29, 2015. The Class Notes belongs to MATH 230 at Yale University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/231011/math-230-yale-university in Mathematics (M) at Yale University.

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Date Created: 10/29/15
Notes on Proofs v 10 by Greg Firiedman September 5 2004 Contents 1 Introduction updated September 17 2004 1 11 What is not in these notes updated September 17 2004 2 12 READ PROOFS READ PROOFS READ PROOFS updated September 17 2004 2 13 Write proofs updated September 17 2004 D 2 General strategies updated September 17 2004 21 Categorize updated September 2 2004 22 Context updated September 17 2004 23 How to proceed updated September 17 2004 231 Know the words 232 Know what you7re trying to prove 233 Know what you7re starting with 234 Build a bridge 24 DRAW PICTURES updated September 17 2004 AgtAgtOOOOOOOOOOOON 3 Writing your proofs Updated September 17 2004 U 4 Some standard proof formats 41 If and only if Updated September 17 2004 42 Checking de nitions Updated September 5 2004 43 Checking cases Added September 17 2004 44 Proofs by contradiction Updated September 17 2004 45 Proofs by the contrapositive updated September 17 2004 46 Proofs by induction Updated September 5 2004 INNCTJCTJCT U CD 5 Speci c Tips 51 Proving that two sets are equal updated September 17 2004 52 Delta epsilon arguments Added September 17 2004 9 1 Introduction updated September 17 2004 These notes are intended to say something useful about how to construct and write proofs They are in an early stage of development I hope to add topics as the course proceeds that are timely to the material So check back often for updates The earlier sections deal more with generally strategies7 but as the notes go on there7s more about particular strategies for particular situations Also7 any feedback would be much appreciated 11 What is not in these notes updated September 17 2004 What these notes will not do is give you a set of instructions for how to prove every possible theorem If such a set of instructions existed7 Id be out of a job7 and Godel would have been a much happier man Even in an introductory course7 there is no xed group of ideas that will get you through everything That being said7 experience leads one to approach certain proofs in certain ways Sometimes the overarching idea is straightforward7 but there are details that need to be checked Other times7 even getting started requires some clever insight So do not expect that reading these notes will imbue you with a level of insight that will conquer all proofs Alas7 such is not to be 12 READ PROOFS READ PROOFS READ PROOFS up dated September 17 2004 While these notes might help you get used to constructing proofs7 there are absolutely no substitutes for exposure and experience You will never get the hang of proofs unless you read them and re read them It is essential to be able to recognize tight logical reasoning7 and the only way to get used to that is by seeing it over and over Read the textbook or other math books to get experience with sound reasoning Then practice picking apart shoddy reasoning in other courses and in the real world You should also read proofs multiple times at different levels I usually start by reading a proof through at a very low level7 making sure that I understand how every line leads into the next You should almost pretend that your checking to make sure there isnt a mistake in going from one line to the next sometimes this isn7t pretend everyone makes mistakes Double check how each equation comes out of the one before it This ensures that I understand all of the little details7 but it does nothing for understanding the big picture After you7ve read at the level of speci c detail7 re read the proof and try to get a better idea of the larger structure Why does the proof head off in this direction How is the main point of the rst paragraph used in the third paragraph What was the original prover thinking when he wrote this proof One almost never starts writing a proof from the details Each time you reread a proof7 you should see new connections between the ideas and get a fuller understanding of THE BIG PICTURE 13 Write proofs updated September 17 2004 It is also essential to get practice Some of this will come from homework problems You should also try to do some of the proof based exercises in the book that aren7t assigned Dont get frustrated if it doesnt all click immediately Rome wasn7t built in a day 2 General strategies updated September 17 2004 Even though there is no magical proof algorithm7 there are some basic skills that will help you approach a proof and some other things that should be kept in mind at all time Let7s run through some of these 21 Categorize updated September 2 2004 This may be obvious7 but it helps to decide at the outset roughly what kind of proof you7re up against ls the statement you7re trying to prove a question about sets ls it about geometry ls it an analysis problem Does it look like it has something to do with calculus Each of these disciplines has its own set of tools and strategies7 and while none of them live in a vacuum7 if you7re asked to prove something about a property of derivatives7 your book on knot theory may not be the best rst place to turn Like everything else about proofs7 it will take experience to begin to recognize the categories of proofs7 but once you learn to make these kinds of classi cations7 it should help you to get started 22 Context updated September 17 2004 This applies more to course work than it does to proving things out in the real world7 but make sure to use some common sense regarding assigned problems If you7re assigned a proof exercise that7s in Section 15 of the book7 chances are that the main idea for how to do that proof is somewhere in Section 15 This isnt 100 infallible advice7 but if you7re stuck7 it should give you a place to start looking for ideas Along the same line7 if you7re asked to prove a minor variation of something that7s already been proven in the book or in class7 dont look to reinvent the wheel Ask yourself how what you7re trying to prove is different from what you7ve already seen proven7 then look at that existing proof and isolate the point where it stops applying to your problem Then see how to modify the old proof so that it ts your new problem 23 How to proceed updated September 17 2004 231 Know the words If you7re asked to prove that the real part of an analytic function is harmonic7 you better making sure you know the meanings of 77real part 7 77analytic and 77harmonic and a review of what a 77function77 is might not hurt either You can7t prove something unless you know what it is you7re proving and what all of the players do That being said7 just writing out all of the de nitions is not a proof either l7ve seen it done Don7t try it l7m not that sloppy a grader 232 Know What you7re trying to prove Know what your goal is7 and dont lose focus lf you7re asked to show that a function has a certain property7 make sure you know what that property is and work back from there 3 Don7t just start throwing words aimlessly down on paper You have a target aim for it 233 Know What you7re starting with Any statement that you7ll try to prove starts with a hypothesis Read that hypothesis twice Know what it says You might also want to write down all of the things that the hypotheses obviously imply Dont get too carried away7 though we7ve got a job to do 234 Build a bridge So now that you know where you7re going to start and end7 your goal is to build that bridge You dont always have to build in one direction Sometimes it makes more sense to start from the hypotheses and work towards the conclusion Sometimes it makes more sense to work backwards7 at least in your initial thinking In the end7 of course7 you7re going to have to take your reader from the hypotheses to the conclusion7 but that doesnt mean you have to think about it in that order Your main job now is to try to remember or invent the ideas that will connect that starting cluster of ideas the hypotheses and their obvious consequences to the ending cluster of ideas the conclusion and the things that obviously imply it Where to these bridge ideas come from Well7 look in the book What theorems do you know that involve one or more of the ideas already in play Can you jump from theorem to theorem What can you ascertain simply by your own reasoning There7s a whole cobweb out there7 and you have to nd the series of strands that get you from the beginning to the end EXAMPLE updated September 172004 Theorem 1 Suppose that f A a B and g B a C are function and that the composition 9 o f is surjeetive onto Prove that g is surjeetive onto Proof 1 WHAT ARE WE TRYING TO PROVE We want to show that given any element 0 E 0 there is some element b E B such that gb 0 Why Because this is the de nition of quotsurjeetive as applied to g E0 WHAT DO WE KNOW We knot that g o f is surjective This tells us that for any 0 E 0 there is an a E A such that gfa 0 9 BUILD THE BRIDGE Given that there is some a E A such that gfa c we want to nd some b E B such that gb 0 But fa 6 B7 and gfa c so we can take b fa in order to nd the b we want Since our choice of c was arbitrary7 we see that we can apply this process to nd such a b for any 0 So we have shown that for any 0 there is some b such that gb 0 Note I was being wordy in the preceding discussion to give you an idea of the thought process When it comes to actually writing the proof7 I would write something like this Let c be an element of G Since 9 o f is surjective7 there exists an element a E A such that gfa 0 Thus there is an element b 6 B7 namely fa7 such that gb 0 So we have shown that for any 0 E 0 there exists a b E B such that gb c ie g is surjective D 24 DRAW PICTURES updated September 17 2004 This topic is so important it gets its own subsection I will say this every day DRAW PICTURES Our geometric intuition is often a lot stronger than our pure abstract reasoning intuition A picture is NOT A PROOF7 but it can be damn helpful in getting the ideas rolling towards a correct proof You will often nds that all of the key ideas in a proof come from thinking about the pictures7 and then the only thing left in writing the proof is translating the pictures into precise mathematical language This last part may sound hard7 but I guarantee you that its a lot easier than trying to think in the precise mathematical language from beginning to end So draw pictures Even of the things that don7t seem like they should warrant pictures And if its too complicated to draw7 close your eyes and try to picture it in your mind I promise this will help Even for the ve dimensional things 3 Writing your proofs Updated September 17 2004 Proofs should be written as clearly and concisely as possible Your ultimate goal should be to convince your reader and perhaps yourself that something is true Make sure to muster your arguments in advance and to write them down in a clear7 logical format If you use a particular de nition7 it may help to state that de nition If you invoke a previous theorem from the course7 you should be clear about which one and what it states When doing so7 it is preferable if you state more than just the number of the theorem from the book While its acceptable to write the proof now follows from Theorem 4377 its much better to write7 the proof now follows from Theorem 437 which says that You should try to avoid writing down facts that are not relevant to the proof A STAN DARD FRESHMAN MISTAKE is to bury the proof in de nitions7 attempting to show the grader that you know what the words all mean While it is important to know your de ni tions7 you should only invoke the ones that are relevant7 and de nitions alone rarely suf ce You must make all of the connections between ideas for your reader IT IS UNAOOEPT ABLE TO ATTEMPT TO HIDE BEHIND AN AVALANCHE OF STATEMENTS THAT YOU DO NOT TIE TOGETHER Eschew obfuscation 4 Some standard proof formats 41 If and only if Updated September 17 2004 These are the proofs that ask you to show that Statement A is true if an only if Statement B is true Of course proving such a thing could fall into any of the following categories7 none7 or more than one7 but I include these in their own section to remind you that in an 77 if and only if7 proof YOU HAVE TO DO TWO THINGS You have to show that Statement A implies Statement B AND that Statement B implies Statement A It is often the case that one of these things is much simpler to do than the other7 but DO NOT FORGET TO DO BOTH 42 Checking de nitions Updated September 5 2004 This happens rarely7 but sometimes a proof is just a matter of checking that something satis es a property straight from the de nition of that property Example Prove that every integer is a rational number Proof By de nition7 a number is a rational number if and only if it is a quotient of integers a fraction Every integer is the quotient of itself with 1 if z E Z then 2 Thus every integer is a rational number See also Example 1 The general approach to these proofs is often straightforward l have to show that every x satis es property l37 though in practice the details may become tricky For example7 if you are asked to show that a function is continuous7 in principle you simply have to check that the de nition for continuity is satis ed by the function In reality7 however7 this may be dif cult to do directly7 and the more useful approach may be to apply other theorems about continuity that have already been developed eg that sums and products of continuous functions are continuous 43 Checking cases Added September 17 2004 Sometimes a proof is just a matter of checking a number of cases although actually checking each case might be dif cult Consider the following example Theorem 2 For any integer z E Z 22 is congruent to U or 1 mod 4 ie its remainder is U or 1 upon division by 4 Proof Any integer z is congruent to either 07 17 27 or 3 mod 47 so it suf ces to check these cases 1 z E 0 mod 4 In this case7 z is a multiple of 47 and thus so is its square So 22 E 0 mod 4 2 z E 1 mod 4 In this case7 z 4n1 for some n7 so 22 16n28n1 44n22n1 Thus 22 E 0 mod 4 3 z E 2 mod 4 In this case7 z 4n 27 so 22 16n2 16n 47 which is a multiple of 4 So 22 E 0 mod 4 litem z E 3 mod 4 In this case 2 4n37 so 22 16n224n9 44n26n21 So 22 E 1 mod 4 D Note that it is important not only to check all of the cases7 but to argue why all possible cases have been considered 44 Proofs by contradiction Updated September 17 2004 Some people dislike this method as it is somewhat indirect In a proof by contradiction we assume that the result is not true and then argue until we get a logical contradiction This demonstrates that it is impossible for the proof to be untrue so it must be true Here is a famous example Theorem The square root of two x2 is irrational Proof Assume that this statement is false ie that x2 is rational That would imply that x2 pq for some integers p and q and we are free to assume that the fraction is reduced p and q have common divisors If this is true then clearly p2 2q2 So p2 is an even number But the only way for p2 to be even is if p is even since the product of two odd number is odd So p 2t for some other integer t and 2q2 p2 4t2 But this implies that 12 2t2 and by making the same argument again we see that q must also be even But now we have arrived at a contradiction since p and q were to have no common divisors which is impossible if they are both even Since our logical was impeccable the only problem must be that we assumed 2 to be rational Hence it must not be CAUTION When doing proofs by contradiction make sure that the contradiction comes only from your assumption that the theorem is false NOT from some mistake that you made along the way 45 Proofs by the contrapositive updated September 17 2004 These are somewhat similar to proofs by contradiction though of a different avor Suppose you have to show that Statement A implies Statement B In other words you want to show that if Statement A is true then Statement B is also true The contrapositive statement to A implies B77 is not B implies not A In other words to prove the contrapositive you show that if Statement B is false then so is Statement A An implication A implies B is true if and only if its contrapositive not B implies not A is time This is not hard to see suppose the contrapositive is true and that A is true Well then if B were false the contrapositive would say that A is false But this is not the case so B must be true Hence if the contrapositive holds so does the initial statement that A implies B I leave it to the reader to show that if the original implication holds then so does its contrapositive Example We showed in class the other day that if f is a differentiable function and a is a local minimum for f then f a 0 We proved this directly using the de nition of the derivative and the de nition of a local minimum However we also could have proceeded by showing that at any point b such that f b 31 0 b cannot be a minimum In the end this approach would also come down to the de nition of the derivative but in general trying the contrapositive might lead to new approached to problems 46 Proofs by induction Updated September 5 2004 Proof by induction is used when you want to simultaneously prove an entire sequence of statements that are indexed by natural numbers For example suppose you want to show 7 that Ells 7 for any 71 Z 1 This is really an entire sequence of statements7 one for each 71 Abstractly7 we sometimes speak of trying to prove the statements 71 for some range of number 71 In the example7 71 is the given statement 21 21 and we want to prove 71 for all 71 E N The basic plan for induction is the following 1 Prove the base cases or cases In other words7 show that 71 is true for the lowest value of 71 Sometimes it might be necessary to prove 71 for the rst few values of 717 depending on the speci c problem E0 Assume an induction hypothesis This means that we now assume that the statement is true for some xed but arbitrary value of 71 This hardly seems allowable7 but l7ll explain below Note that when I say arbitrary7 I mean arbitrary We dont assume here that the statements are true for 71 up to 5 we assume they are true up to N 7 1 Just like that N could be anything 9 The induction step Show that the induction hypothesis implies the next case In other words7 if we assume by induction hypothesis that 71 is true for N 7 17 then in this step we show that N is true7 under that assumption 7 That7s it We7re done If you7ve completed the above process7 you7ve shown that 71 is true for all 71 So whats going on here7 and why can we make that induction hypothesis The easiest way to explain is to work backwards I claim that if you7ve followed the above steps7 then 71 is true for all 71 So lets pick a case7 say 7 Why is 7 true Well7 we showed in Step 3 that 7 will be true if 6 is true Similarly7 that same step7 show that 6 is true if 5 And so on Eventually we get down to7 say7 2 will be true if 1 was true But if 1 was the base case that we treated in the rst step7 we know that its true So the 2 is true7 and 3 is trueand 7 is true Let7s show how this works for our example H Prove the base cases or cases In our example 1 say that Zia k 1 But Zia k 1 So this is true We have established the base case 3 Assume an induction hypothesis Here we just say that we assume that k W w is true That7s it 9 The induction step Okay7 now show N7 using our assumption about N 7 1 We have N N71 2k kN k1 k1 N721NN N27N2N 7 2 2 7N2N 7 2 7NN1 7 2 Note that from the rst line to the second line7 we used the induction hypothesis to N71 i N71N replace E k1 k With f 4 That7s it were done The theorem now follows by induction One also sometimes uses generalized induction In this case7 instead ofjust assuming N 71 is true in the induction step7 we instead assume that 71 is true for all n S N 7 1 and use this to proof that N is true This also works7 essentially for the same reasons NOTE Dont be too xed about notation For example it is often notationally more convenient in the induction step to assume N and use it to show N 1 The notation is slightly different7 but clearly the idea is the same Similarly7 it may also be necessary to prove a few base cases in order to get the induction going 5 Speci c Tips 51 Proving that two sets are equal updated September 17 2004 If you are asked to show that A B7 where A and B are two sets7 the simplest approach is usually to show that A C B and B C A In other words7 show that every element of A is also in B and that every element in B is also in A This implies that the elements of A and B are the same7 ie that the sets are equal Dont forget to do both parts 52 Deltaepsilon arguments Added September 17 2004 These are perhaps the most challenging arguments for students to become acquainted with For one thing7 there are at least two quanti ers a for all and a there exists Let7s look again at the de nition of continuity of a function f at 1 De nition 1 The function f U C R 7 R is continuous at a if for all 6 gt 0 there mists a 6 gt 0 such that ix 7 al lt 6 implies 7 fal lt 6 Lets just look at the rst part of this for all 6 gt 0 there ewists a 6 gt 0 such that77 This says that if choose any 6 gt 0 then there must be some 6 gt 0 that makes the following statement true Note that 6 MAY DEFEND UPON 67 but the 6 is free to be arbitrary In this speci c de nition7 the concept is as follows we want to know that if z is close to a then f is close to fa The above de nition provides a rigorous mathematical de nition of what we mean by close Sometimes this is called a challenge procedure if you challenge me to make f 6 close to fa then I tell you that it can me done so long as we stick to points that are F close77 to a This is just a sample delta epsilon de nition7 and of course there7s nothing to prove about a de nition it is what it is The actual delta epsilon arguments you will have to make come about in trying to apply the de nitions So lets look at the following theorem7 whose proof is7 in principle7 just a matter of checking the de nition7 though of course the actual checking itself requires a little ingenuity Theorem 3 Suppose that f andg are both functions R 7 R and that both are continuous at 1 Then the function f g is continuous at a Proof Well7 we need to check that f 9 satis es the de nition of continuity at a we must show that for any 6 gt 0 there is some 6 gt 0 such that lf g 7 f g al lt 6 whenever lx 7 al lt 6 So7 let us pick an arbitrary xed 6 gt 0 If we can nd a 6 that works for this 67 we will be done even though 6 is xed for the moment7 it was xed arbitrarily7 so whatever argument we give would work for any 6 Okay7 so now that 6 is xed7 how do we go from here Well7 we want to show something about w 9W 7 f gal me 9a e fa e 9al Since we already knew something about the continuity of f and g at 17 we expect that to come in7 so lets rewrite this as 7 fa g 7 gal7 which looks a little more like it has to do with the continuity formulas for f and g In fact7 if we want to use those formulas7 we should try to compare this to something involving 7 fal and 7 gal For this we can use the triangle inequality to write g 7 fa 7 gal S 7 fal 7 gal This is looking good7 since continuity of f and 9 should allow use to conclude that the right hand side of this is small In fact7 since we want to make g 7 fa 7 gal less than 67 it will be good enough to show that each of 7 fal and 7 9a is less than 62 right Now using the de nition of continuity at 1 applied to each of f and 9 we know that there exists some of gt 0 such that 7 fal lt 62 and 69 gt 0 such that 7 gal lt 62 Note you may be thinking7 Hey The de nition of continuity for f and g tells us something about 67 not 62 7 but remember the 6 in each de nition is arbitrary it just represents any number gt 0 We know from the de nitions that there are appropriate 6s for any number gt 07 including 62 for the current xed 67 so the de nition applies and allows us to conclude there are the 6s we want Okay7 so now we have two deltas7 of and 69 each one giving us a radius around a such that the corresponding function f or 9 do what we want near 1 But in order to get lf 7 f gal lt 67 we need BOTH of these to hold So we take 6 min5f597 the minimum of the two So if lx 7 al lt 6 then lx 7 al is smaller than both of and 69 So if 10 lx 7 al lt 6 then 7 fal lt 62 and 7gal lt 62 So by our triangule inequality argurnent lf 7 f gal lt 6 Thus we have found a 6 that works for this 6 and we are done 1 There is another truth that is often handy when dealing with delta epsilon proofs and this is proven in the text In choosing 6 it actually suf ces to nd a 6 that yields any function of 6 that goes to 0 as 6 goes to 0 To explain what this means consider again the previous example We chose 6f and 69 so that 7 fal and 7 gal would be less than 62 Suppose instead that we had chosen 6f and 69 only so that these distances would be lt 6 Then if we de ne 6 rnin6f6g again we will only be able to conclude that for lx 7 al lt 6 then lf 7 f gal lt 26 Our rst thought is Oh no It didnt work77 But this nice theorem in the book says that it actually did since 26 7 0 as 6 7 0 THIS IS SUFFICIENT This nice fact often comes in handy as it is often much trickier to get the calculations to come out exactly to 6 The theorem tells us that we could make this happen with some extra work but we dont need to


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