The important experiments in the field of biochemistry and genetics laid down the foundation of molecular biology. The most important experiments among those were the description of DNA structure by Watson and Crick and in the field of genetics, the main experiments which contributed towards the development of molecular biology were the experiments of Avery who proved that DNA is the genetic material. So these experiments, they actually laid down the foundation of molecular biology. Now the molecular biology was born and the new discoveries were being made in the field of molecular biology. So the question was how DNA is expressed, how it transfers its information in the form of proteins. So this problem was solved in 1957 by Francis Crick who laid out the central dogma of molecular biology and this central dogma foretold the relationship between DNA, RNA and protein. It described how DNA transfers its information to RNA and then this information is translated in the form of proteins. So here in this illustration you can see the DNA molecule. The information in the DNA molecule is copied to the mRNA molecule and then this information is translated from mRNA to the proteins. So the information which is present in DNA molecule is exactly copied in the form of protein molecule. So in this way this information is carried from the nucleus from the DNA molecule and it is transmitted to the site of protein synthesis where this information is translated in the form of protein synthesis. Next important discovery was made in 1958 by Matthew Macilsen and Franklin Stahl. They proved that the replication in a DNA molecule was semi-conservative because this was another important problem that how a DNA molecule is replicated into two molecules and then into four molecules. So in this picture you can clearly see this is a DNA molecule and when it replicates one strand goes to this molecule and the other strand goes to this molecule. So both the blue strands here and here they are parental strains. They have come from the parental molecule of DNA and the pink strands they are synthesized new. So this is called semi-conservative that is one strand is conserved. In the next replication again from this molecule one strand is conserved and the other strand is newly synthesized. So this is called semi-conservative replication and this was described by Matthew Macilsen and Franklin Stahl. Next important contribution was made by Marshall Nirenberg and Gobind Khurana. They work independently and they cracked the code in early 1960s. They found that three bases constitute a code word called a codon that stands for one amino acid. It means the three words on the strand of DNA they code for one amino acid. So in this way all the information for the synthesis of proteins containing 20 different types of amino acids is present in the DNA molecule in the form of codons. Next in 1970s the important achievement was gene cloning. Scientists have learned to isolate genes from some organisms, place them in new organisms and reproduce them by a set of techniques which is called gene cloning. So in this way the genes, the characteristics from one organism can be transferred to the other organism. It was an important achievement in the field of molecular biology and it gave rise to the genetic engineering. So with the help of this gene cloning and genetic engineering many genetically modified organisms were created and these organisms included plants and animals and many of these genetically modified organisms are in practice, they are in the field and still many new are being created. So the new organisms, genetically modified organisms they have desirable characteristics. As you desire you can produce those characters in the GMOs. The important contribution, perhaps the most important contribution in the field of molecular biology is the human genome project. The human genome project was launched in USA in 1990 and was completed in 2003. The main goal of human genome project was to sequence the whole genome of the human and to identify and to map all the genes present in the human genome. There are approximately 35,000 genes present in the complete human genome. Although this human genome project it was successful to the extent that it sequenced all the genome of human beings and it mapped all the genes present in this genome but many of the genes are still undescribed. That is the functions of those genes is still to be described.


You're still early in the game. What do you want your legacy to be when it comes to this game, man? Be the system. The system? What's the system? A boxing. Be the system of boxing, life. Just be the system. Be to spread my wings and having my people next to me spread their wings. That's it. I ain't really got no big goal of this sports and all that. I just want to get my money. And just be more myself.


In uniform circular motion, we have a particle that follows a circular path at a constant speed. We have two ways of describing that motion. One is with linear variables and one is with rotation variables. For the linear variables, the position is given with the power along the curve, the power along the circle. Which, of course, is its linear speed as the time increases to the next line, the linear speed, being the derivation of that position. We can also give it as a rotational entity along the circle, which should be equal to the rotational speed omega times time. Omega being the derivation of the angle of time. Now, if we consider the time it takes to go around one time as the period capital T, we can say, oh, okay, so this is my linear speed is constant. I have to go around once, so a total path of 2 pi r, one second few, is over the period. Which would give me the linear speed similar for the angular speed. So, okay, go around once. It's 2 pi in radians over T. So, as you can guess, we're going to measure the angular speed in rats per second. We angle in rats for the linear distance with the meters, and then we get the linear speed as the linear speed. Now, if we compare those two times here, is there a direct connection between the linear speed and the angular speed? Yes, there is. See, we have 2 pi r over T. Here, we have 2 pi r over T. That means we can calculate the linear speed as the angular speed times the radius. This has worked out with the units. If you look at it, that's per second times the radius in meters. You see that? And the radius per second. The radius is one of the units that can appear and disappear. So, again, yes, I do that meters per second. Now, the next thing that I want to look at is the linear velocity, which is a vector. So, I will have to describe it in i-hat and j-hat protection. So, first, I do have this e v times... And if I look at my x component of the velocity, when I'm here at zero x component, at 90 degrees, I have my negative moment. So, it's component at 180 at zero again. And at 270, I have my highest possible value. So, if you think about it, you'll have a function to do that. And it will come up that it is sine of the angle that does this, the angle which we say is omega t. However, at 90 degrees, we are at the negative value. It's in fact a minus. And for the value at zero degrees, I'm doing it with my highest speed. Then we have zero speed at 90. Negative speed at 180. And the closer at zero speed, again, in my direction. Once we're at 270, so plus cosine. Because now I can try to figure out what does the linear acceleration look like. The acceleration, the linear acceleration, is the derivation of the velocity. So, if I derive this equation to that, I should get an expression in i hat and j hat for my linear acceleration. So, let's do this. v is the main just constant. If I derive sine, I get cosine. Don't forget, in the derivation, transform it up. Plus, if I derive cosine, what do I get? I get minus sine j hat. Now, let's rewrite this a bit nicer. I could say, okay, this is my v, which is omega r. The omega that I have here and here. So, omega square r minus cosine omega t is sine of t j hat. So, let's have a look at what that actually would mean. My acceleration has a maximum amplitude of omega square times r, which we disappointed. If I look here at my position here, I would have cosine omega t. So, minus one x and sine of zero. So, here my acceleration is on this one. What about here? At 90 degrees, I will have zero for the x component. Minus one for the y. If I'm here, I have minus cosine omega t. So, to plug in 180, I will get positive x component. And again, zero y. And down here, it will be at 270 degrees. Zero x and plus one for the y. So, the acceleration that we have is always pointing to the center of the circle. This is also what we call this centripetal acceleration, of which the magnitude must be omega square times r, which if I plug in this equation here for omega, gives me e squared over r squared. So, that will give me...


guys are doing the full sellout promo today is February 19th 23 we're giving off $100 in break credit to the top three participants we have 504 participants in this again this is from all of the breaks that we did today if you bought a spot in tonight's breaks the evening breaks with me we are doing a you have a chance at $100 in break credit so good luck guys here again this will be added to your account tonight gonna roll the dice we're gonna go 10 times 10 times 1 2 3 4 5 6 7 8 9 and 10 good luck everybody that is gonna be Jerome Z in prism first off the line Bernie S and impeccable Don Russ 13 and William M and flawless 27 random left side serial number so congrats to Jerome Bernie and William thank you guys again thank you everybody for getting tonight sold out we'll start personals in just a minute here do one thing really quick here again guys absolutely $100 in break credit what he points is that right guys gonna add these points in just a minute let's get going or let me go grab personals really quick and we'll get rock and roll in here we've got Eric with nine or eight bowman draft sapphire and guys not gonna get to every every break today so I do apologize all right beat right back everybody congrats hang saw the Warriors now it's like oh coming your moody that'll be cool nope no sir not at all we'll get that magged up for you hang up of course show hey oh tiny triple patch two of three angels very nice eight RBI 13 K that is a sick patch in the center to Cody Bellinger boom at a West I saw on there that nasty patch on that thing and tiktok we got our first box that is a goal wow that is so sick oh my god green another one we've got sky more one the chiefs congrats buddy rookie excellence one-on-one all right guys we have a big card here we have a big card here letter plus auto relic book card oh my god one of one David Ortiz kaboom red socks with a monster booklet hits come on man are you serious John Q Red Sox white whale David Ortiz silver slugger triple patch auto crinning plate one of one 10 of 10 show hey oh tiny massive hit right there buddy let's go that's so sick that'll go straight into a mag wow wow wow wow oh that is a that's a luka sushi image variation orange rookie five of 25 big card I've never pull I've a pull one of these yeah the variations are sick that Tampa Bay now that's not oh got one already let's go that I am so jealous of that Jackie Robinson Corbin Burns there you go first one of the day one of one Craig a BGO that's my favorite player of all time right there literally that is so sick that's those 2000s Astros jerseys to wow he's got Houston congrats H town fracture in a gold wave F1 racer rookie Joe Guan Yu that's really nice these inserts are really cool love the color on that that's sick but Red Sox legends here going back to Regina triple auto four out of the devours wow sick and Grimes rookie and not expecting that stuff curry jersey number wow holy cow there's the 30 I saw the Warriors and I was like oh comminger moody that'll be cool nope no sir not at all we'll get that magged up for you hang up of course show hey oh tiny triple patch two of three angels very nice 8rbi 13k alright guys we're back me upload that full sellout promo really quick


Today, we are going to start a new module, where we would be focusing exclusively on how our own emotions, they are praised by us, how do we experience our emotion, but that is not our prime focus. Our primary focus is that, how our emotions play a role in making ourselves more and more adjusted. What we would do is that, we would initially begin with the basic concept of emotion. Those of you who have been to the introductory psychology course, you must have read emotions at length there. Here, I would know for the benefit of those who have not been into the introductory psychology course, what I would do is, very quickly I will summarize the main thirst in this area, the knowledge that exists, very, very crisp type of, very quick type of a summary. And then, we would move towards understanding emotions with respect to what is needed by the society, what is needed by the environment. How do we experience it and why is it that we experience certain type of an emotion in a given type of situation, based on what we think of it. That is the whole appraisal factor in emotion. And we would primarily take the goal congruent emotions and the goal incongruent emotions. We will basically know the basic emotions, we will bifurcate them into two clusters. And then, for each of them, we would see that, what is the primary appraisal process, what is the secondary appraisal process and why finally, something makes us happy, something makes us jealous, something makes us guilty, that would be the focus. And for each of them, we would then try to understand that, you see that my happiness has made me more and more adjusted, given the type of situation I was experiencing. So, this is the whole focus in this very motive. I am just taking one of the definitions of emotions, given by Isaac, where he says that emotion is basically a complex state, which involves heightened perception of an object or a situation, widespread bodily changes and appraisal of felt attraction or repulsion and behavior organized towards approach or withdrawal. Now, frankly speaking, many many things are there in this definition. What we will do is, that we will break it up for our convenience. And this will very easily let us know, that what are the basic elements in of a emotional state. First it says that, it is a complex state, which involves increased heightened perception of object or a situation. So, there is an object in your environment or there is a situation, that you are experiencing. And the perception of the object or the scenario, it is relatively heightened in nature 1, 2, that widespread bodily changes could be induced, because of the presence of the object or the situation, where you are. So, there are certain physiological changes, that takes place to third, that an appraisal of felt attraction of repulsion. Either that very bodily changes and the object or the situation attracts you, you move you feel moving towards it or it induces a feeling of repulsion in you, where in you feel going away from that object or that very situation. And the fourth, that you organize your behavior, which either goes for an approach type of reaction or for a withdrawal type of reaction. So, either you tend to know go and face that object, that situation or you tend to withdraw from that object or from that situation. So, primarily four important processes are put together in this very definition. Now, if you look at the basic elements of emotional state, it has multiple processes involved into it, first the physiological reaction. Physiological reaction would primarily be, that the moment you start experiencing an emotion, certain changes takes place in within the physiological system of the body. I would slightly deviate in terms of making you understand, what type of physiological changes takes place. There are interesting theories in psychology, in fact the very first theory of emotion, that was proposed long back. It talked about the fact, that say you respond in a given situation, you have a stimulus, a situation, an object, you respond to it and it is your response that makes you experience the emotion.  So, primarily it is not the presence of the dog, who is trying to reach you, makes you scared, rather it is your own physiological reaction that makes you scared. Little later this theory was revised, which again said that these two processes run parallel. And since they know many, many, many things have been added to the research in emotion. But something that remains universally true is, that certain set of emotions they induce certain changes in the body and they would elevate the processes of the body. Say for example, the blood pressure might increase, the pulse rate might undergo a change, the sensitivity level, the electrical conductance at the surface of the skin that might undergo a change. In fact, there might be certain changes that could be observed in terms of say the brain waves. There are interesting studies, which also says know that what type of emotion one has experienced can also be detected only by looking at the contraction of selected muscles on the face. There are very good EMG studies, the electromyrography study, know where this very area of the face, know sensors are attached to it and you just look at the object. And depending on the changes that takes place in terms of contraction of these muscles or rarifaction of these muscles, one can make out what type of emotion you actually experienced. This type of changes that takes place within the body is basically one of the primary indicators, which is used in lie detection tests. So, in cases of say forensic investigation, if somebody is put under lie detection, I am sure many of you must be aware of how lie detection is conducted. Initially what happens is say if you are a suspect, one investigating officer know interrogates you. That investigating officer would be somebody from the local police station and the record of the question that was asked to you and your responses are retained by the local authorities. When you are taken to a forensic science lab, those questions along with know many other questions also, in a random order they are asked to you. Once again, all sensitive indicators of the bodies are taken like blood pressure, like pulse rate, like know the perspiration rate, the know change, GSR changes the change at the skin conductance level. It is not very frequently used, but there could be a possibility wherein you can even analyze primarily two of the brain waves to find out if there was some type of a deception here in this case or not. And very in interesting research, which is being conducted nowadays is in terms of the pupillary dilation. So, just looking at know how much you dilate your pupil could also be constitute as an indicator of deception. Now, certain questions which does not invoke some type of emotion in you and therefore, your physiological reactions are different and suddenly there is a change in the physiological reaction when some uncomfortable questions are asked to you. Now, the changes that a lie detector records on the basis of which the scientist in the forensic science lab declared that yes, you did speak lie on these questions are again based on the physiological reactions, which are recorded by the given machine. Same technology is also used when you look at your own physiological reactions in a biofeedback clinic. To the best of my knowledge, we do not have too many biofeedback clinics in the hospitals in our country, but I know selected hospitals where there is a specialized clinic called biofeedback clinic. And biofeedback clinic is nothing, but basically you are simply made to see your bodily functions autonomic nervous system functions what we are referring to here as physiological reactions on this screen.  And all you are told is that see there is a by larger standard template how people should ideally breathe or what is the standard way the heart should beat, either your heart is beating slowly or it beats faster. You just look at the monitor and try your best to make your own physiological functions match the standard template. And believe me few minutes of trial and you would realize that yes you can manipulate it, it is not at all difficult. Now, few minutes of exposure and you realize oh I can know tamper my own bodily system then. This means that if you are periodically asked to come to such clinics and using that machine you are asked at say you know usually this is know how your physiological function works this is how your body reacts. And now with the help of the machine you have learned how to tame your own bodily system after certain trials you do not even need a machine. You know that fine this is what I think you know that this is what I feel this is what I think and this is how I get a control over my own physiological reaction and it is possible perfectly possible. I do not think we will have time to do that, but if you visit the psychology lab here we also have this machine and you can see for yourself know how easily you know you can manipulate your own physiological reactions. But that was in terms of control mechanism. Right now what we are talking about is that one of the prominent process that is involved in emotional reaction is changes that takes place you know in a covert fashion in terms of the physiological functions of the body. So, whether you like it or not certain types of emotion will lead to certain type of changes within the physiological mechanism. Do there would certainly be certain behavioral reaction when you experience an emotion and that is you know get affected with the presence or the absence of a particular type of an object or a situation the initial definition that we looked at. So, presence of the stimuli in the environment makes you behave in a particular way. Think of certain situations right now when I am quoting it as example it might appear hypothetical, but all of you I am sure must have experienced such type of things.   In the same train you enter there is no foul smell you take your seat and then you realize there is a beautiful girl on the other side of the seat your entire you know behavioral reaction changes.  So, irrespective of you know the fact that you remain constant as an individual the moment the situation would change in the environment depending on the person environment relationship you would realize that you would be tempted to come forward with a behavioral reaction. Third important factor in emotional processes is the expressive reactions. Expressive reaction once again is a is a byproduct of the person environment interaction where you express yourself. Now, in psychology you will find you know very rich literature on both types of expressions verbal expressions and non-verbal expressions verbal expressions where you know verbalize your feelings non-verbal expressions it could be through your facial expressions it could be through your gestures there could even be you know imagination thought dependent type of a reaction that you contemplate. Now, expressive reaction usually would be those you know expressions which are very overtly visible, but there is always a possibility of non-verbal reactions also. This is again you know great deal of research on all of this you know verbal non-verbal all of this has been very extensively researched. And then the last one is that there is something called subjective experience. Now, subjective experience is an experience that you as an individual had in a given type of a situation. And the experience that you had might not match with rest of the people who were in the same situation it could be extremely extremely different completely divergent from others who had you know experience the same thing, but in terms of subjective experience you find it there could be a difference. For example, the same example if we take you confront say an object that makes you very scared and the expression of fear that you have in your face might be different from the expression of fear that somebody else has on his or her face. The intensity of the expression might change the way you respond to that scary situation it might be different and it could be completely different know when a third person shows a reaction to a scary situation. In terms of behavioral reactions we all differ in terms of subjective experience also we differ, but then there is a great deal of what you call overlap when it comes to expressive reaction and when it comes to physiological reactions. So, these are interesting know developments when you look at emotion as a process many things have been know studied at length. For example, how the physiological systems know get affected in a given emotional state 1, 2 the changes know that we were talking about in terms of the bodily reaction when it comes to forensic investigation and other types of changes. Even the recent trend of work which is you know going on in terms of looking at the pupillary changes and if you again look at the current set of theories that are being proposed in this area you find a beautiful description of something called cognitive emotion regulation which primarily means that you can have some type of a cognitive appraisal based control over your own emotional processes. Think of a situation you are an office bearer who feels something, but is not supposed to express it. However, strong you feel within outwardly you cannot express it outwardly you have to still remain very quiet composed. I will give you some real life examples minus the names. I know of somebody who was once caught in ambush in central part of India while he was returning back after a search operation in a nuxle affected village. So, the whole convoy was moving and that was the joint team of the local police force and the central reserve police force. He being the officer was in one of the gypsies and the vehicle the patrol vehicle which was just in front of him which was supposed to know clear of the route for these people suddenly landed up on a mine and then it blasted and because his vehicle was just behind it he could see everything. So, he saw the vehicle now flying up in the air then body parts getting mutilated falling down on the ground. And he told me that two of his bodyguards who were with him for last several years were also in that vehicle which exploded. And then say for example, if you know some somebody called Ram Singh this is not the true name hypothetical name. Now, how difficult how painful it is to see Ram Singh getting converted into pieces. It agitates you within but then say a media team approaches you asks you about your experience. And you are supposed to show heightened degree of calm and composer on your face that yes this is what happened and we will take stringent action. Inwardly you feel you know that pain outwardly you still have to show that no no no I am a brave soldier who experienced it and we will fight back I am not at all disturbed. This is one of the examples when I am sure know sometime some situation might have confronted you when you must have felt crying or when you must have thought of enough bursting into anger. But then because of situational destructions you decided not to burst because of situational interpretation you decided not to cry there. So, these are know the part of cognitive emotion regulation we are not going into the details of it. In terms of expressive reaction once again know it is important for us as an individual not only to display through our expressions but also we should be skillful in terms of understanding the expressions of others. Again a very true example a very different type of a situation somebody known to our professional group lost one of the family members. And then we decided to visit his house pay our tribute show our solidarity to that family. What we had decided was that all of us will go independently.  In our group we had somebody who was of a little advanced age but was not married this has nothing to do with marriage. But what I am trying to say is that perhaps that very person did not get those many opportunities in life to know judge expressions of others repeatedly when you are in a group setup when you live in a group setup you periodically look at the expressions of others. And therefore your ability to know decipher the emotions of others through the phases increases. Now this person goes there and instead of you know asking an appropriate question simply ask so how are you. Now somebody has lost somebody in the family and you asked that person how are you. Now socially this is an irrelevant question know should not be asked at that point in time. And rest everybody started looking at that very person know that what type of question did you ask. So in terms of know adjustment it is also important that you tend to mask certain things you tend to neutralize certain things we will come to those things little later. When you feel angry but on the face you try you to know retain that neutrality as far as possible. You do not show your expression just you gave your mid semester examination and say for example one of the questions you did not understand appropriately. Fantasy, motion, independence, dependence you come and ask me and I say that I cannot explain it right now. And may be that you felt know giving question paper back to throwing that question paper back to me take your question paper now I am also not interested giving the medicine exam. I have asked for you for a clarification and you say no during examination I will not explain it. But then you do not have a choice because you know that you run a higher risk if you showed this type of an emotion.  But I do not have a choice but to write the answer to the question and know it saves us a lot because know the moment you look at expressions of others you realize that find what the other person experiencing and accordingly you can tame your own reactions. And therefore you try to minimize conflicts as far as possible. Now behavioral reactions what we have talked here was know that physiological behavioral expressive and subjective experiences. Now we will drop three of them we are now exclusively looking at the behavioral reaction because we are into the psychology class right now. Had this been a physiological psychology class then we would have gone to the first one first know the physiological reactions. But we are primarily now banking on the behavioral reactions and since the slide we would only and only talking about the behavioral reactions. Now behavioral reactions they directly govern our approach or avoidance reactions. Now I decide to approach you I decide to go away from you and that primarily know decides in the long run know how would be the relationship what type of relationship finally we would maintain. So are we going to have a stronger bond between the two of us or is it that we would have a very weak bond between two of us or is it that now onwards we are strangers and we will not interact at all. And primarily it is the behavioral reaction which also would decide in the long run the adjustment level of both the people who were put in that very situation.  Somebody with whom you had a terrible fight and then you understood that there is nothing so beautiful in retaining the fight. But there is a beauty in understanding each other and becoming friends and you decide to become friends rather becoming enemies.  But I do not appreciate your view point and you do not appreciate my view point and hence we remain separated completely insulated from each other. We will not interact with each other at all. All such possibilities exist. So basically these behavioral reactions will primarily decide whether you would approach that individual or you would try to withdraw from that individual. It could be for a given situation also whether you would approach certain type of situation or whether you would try to stay away from certain type of situations. Later on when we come to that self-assessment session that I had promised as part of this course. At that time we would also be you would yourself would self-administer a test which is primarily designed to measure social anxiety in you. And one of the components of social anxiety is the fear of negative evaluation. I as an individual feel I will not go in that situation because people would start judging me. Say if I ask you anybody who would like to share one experience here many of you would hesitate coming to this side of the desk. The reason being when you are on that side you feel still it is a safer place for you. The moment you come here you realize that everybody is looking at me. There is a default assessment that takes place. And you do not want to be under scrutiny by so many people and therefore you say even though you want to say something you will keep quiet after the lecture is over you quietly come and ask the question. This is a very common practice because you keep thinking twice thrice how to ask a question in front of others because others will start thinking array you could not even understand this also. I understood it long back before coming to the class. So, this is important behavioral reactions because the whole of withdrawal and avoidance the whole of the reaction in terms of whether you would approach whether you would avoid in the coming days that would be decided by your behavioral reactions in the long run. Coming back to emotion since the time of Darwin there has been long debate in terms of identifying what are the basic emotions or is there a component of universality in emotion. This is a very old debate now it has been resolved thankfully. Basic emotion means emotions which by default will stay in all human beings. Universality of emotion means whether you are here or in some other part of the globe but you would certainly display your emotions this way. So, that would be the universality and there were many models. Now, the understanding is that we have six basic emotions as human beings. So, whether you are in India whether you are in some other country or even if you have been born at a very isolated place still you would have know these basic emotions. Happiness, sadness, fear, anger, surprise and disgust these are our six basic emotions. Happiness, sadness, fear, anger, surprise, disgust fine. Rest of the emotions they are considered to be a combination of two of them. There are few more emotional experiences but they are considered to be the intensification of these basic emotions or they are supposed to be the by product of combination of two of these basic emotions. We would not go into those composite emotions we would not go into those you know intensified versions of emotional basic emotions. There is a very beautiful theory given by Plutchik where he talks about all this you know where he talks about the folding it is like say unfolding of the petals. So, say if you have a leaf of a flower you unfold it once you unfold it twice. So, he has taken all basic emotions first level of intensified reaction second level of intensified reaction and. So, you have the whole sphere and then he also talks about combinations when x 1 and x 2 combines then what happens when x 1 and x 3 combines then what happens x 2 and x 3 then what happens beautiful beautiful theory but we are not right now going into those theories. Another important way of looking at emotion was in terms of certain major attributes major attributes like valency of the emotion. So, we have six basic emotions now talk in terms of valency of emotion valency would be you divide emotion into two clusters positive emotions and negative emotions happiness is a positive emotion surprise is a positive emotion sadness fear anger discuss they are all negative emotions. So, one way of looking at emotion could be the valency of emotion second important way of looking at emotion could be the directionality of emotion directionality means whether that emotion makes you approach or avoid the situation happiness what type of emotion is it it is an approach emotion I feel happy being here therefore, I would come at 8 o clock tomorrow also approach emotion this type of situation makes you very sad come on here you try to stay away from that situation. Happiness approach sadness will repel you are angry is also an approach emotion you feel angry and then you feel directly going to the source of anger and express it in the full blown shape if you calculate that the risk is worth taking if you calculate that the risk is not worth taking then instead of you know reacting to that particular individual you show some other form of aggressive retaliation what forms that we would come after two modules when we will exclusively talk about aggression and how it affects adjustment. So, valency directionality and again if you see here directionality the approach and avoidance once again comes here. So, behavioral reaction will always have this thing by default there is a very interesting no proposition it is called eco cultural framework given by Barry and later on revised in 2002 by Barry and his colleagues which talks about three kinds of incidental influences sorry three kinds of and antecedental influences first the ecological indices to the socio-political indices and third the aggregate psychological characteristics. Once again we are not going into any of these theories any of these framework, but this is just to say know that you still have certain major factors which influences your behavioral reactions in a given situation and these behavioral reactions will in turn trigger a set of reactions the other person in the environment will react back to you this in turn will demand you to react in certain way and this chain might get triggered this could be one possibility. The other possibility is that your reaction meets some type of buffer reaction wherein your native reaction gets diffused and therefore there is no retaliation from the other end and you are very easily accommodated with all type of behavioral reactions that you have shown. Now, looking at emotion and adjustment together the relationship between you know emotion and adjustment in terms of how emotion helps you adjust in a given situation and also in terms of how it finally facilitates your subjective well being it has also been very well studied in behavioral sciences. The quality of adaptation shown by an individual is proportionate to the happiness one derives in life and overall life satisfaction. This means that your own qualification as an individual to adapt to a given type of situation that is directly proportionate to how much of satisfaction and happiness you would derive in life. I think last week beautiful piece of research has come forward which says that you must have heard the famous proverb that money cannot buy you happiness famous proverb. Banking on this the researchers based on their empirical data says that money cannot buy you happiness directly, but with the help of money if you buy things for others that in turn can make you happy they got the point. It is a very interesting research know that you have money you can afford to buy things for others and hence you buy things for others and you derive happiness out of it. So, money has not led you to buy happiness directly, but indirectly you can still derive happiness. It is something like say you buy crackers and you know burst them you derive happiness out of it. The second possibility where you buy crackers and some of the crackers you give it to others who do not have crackers with them and you watch them now blowing it and their happiness in turn induces sense of happiness within you. I am sure at railway stations it is a very common site you buy some food stuff for you and somebody comes asking you begging you for food. I am sure all of you must have seen that if you have traveled by train and know many times that urge comes from within know that you just took the plate and here is somebody know with folded hand ask you for food and instead of taking food yourself you give the whole plate to that individual you buy another plate for yourself. And the happiness that you derive out of it is know far greater than the satisfaction you would have otherwise derived while consuming that food yourself and saying that this go from here.  But I will take one example of each of them, but what happens that this know adaptation finally makes others and you yourself realize that fine I have proven my know ability as an individual to appropriately respond in this situation people would appreciate you would also feel satisfied.  And your request does not fetch you the desired result know you tell them that this is a result seat for me and the same. So, sit somewhere the person who is occupying your seat does not leave that very seat it is also a very common type of a thing know in certain parts of this country and suddenly you intensify your emotion. You burst and the person who is occupying your seat realizes that there is no point still retaining the seat. So, let me vacate it says now you take your seat else, but there is no point showing anger like this now. I was just sitting because you did not come and then he leaves the seat you get your desired outcome your outcome is based on the intensification of the emotion. Second type of situation where you realize that there is a need to neutralize your emotion the person occupying your seat seems to be much billwilt has couple of arms and ammunitions with him can show you know all types of things that you would otherwise never think of experiencing in life.  You decide not to show your anger and rage because your calculation says that you yourself might be at the recipient and if there is a fight between two of you. Therefore, you quietly occupy a seat which is next to it you do not argue and there could be third situation where you mask your emotion masking of emotion means the inward emotion is something else, but outwardly you put some other emotion over it you look at the person say you are sitting please no problem you can sit here I book sits for others only you please keep sitting here. So, you did not even think of neutralizing your feeling rather you thought of masking it putting another emotion over your original feeling. We do all these things in our life at times we show our proportionate reactions at times we intensify our reactions at times we mask our reactions at times we neutralize our emotions because we realize that in this given situation this is what is prudent this is what will give me the best result and therefore, you do it and because you have calculated that this will give you the best possible result and you get back therefore, you derive happiness out of it that I finally got what I wanted you derive a sense of satisfaction and once again this all this feeling of happiness and satisfaction that you derive out of your ability to adapt in a given situation it positively affects your subjective well-being your own feeling of how good you are how happy you are that experience increases. Now, emotion has certain biological universals, but what makes it significant for adjustment process is the socio-cultural variability. So, there are biological universals know that this emotion will have this this emotion will have this, but then what you realize that there are variation in terms of the socio-cultural demands and it is basically the variance in the socio-cultural needs in given situations which finally, puts you under some type of pressure that your emotion needs to be like this.  Now, few societies such as the agricultural society they lay emphasis on the compliance whereas, other societies know might cherish independence know. For example, our society requires more and more of compliance where major say stakeholder stakeholder who has otherwise more power and authority invested in him or her decide certain things and then people try to comply to it you go to a shopping mall and then the wife says that why do not we have the kids says that why do not we go to those know through the fun zone where we can have some toys to play and you as the major stakeholder in the family says no we will all go to this you take the third direction and you realize that very quickly the other members of the family will comply to you. The wife might say how irritating this person is I feel having bhel poody and he is not even allowing me to have it. The kids might feel that how boring my father is that he is not even allowing us to play in the fun zone, but the fact remains that both of them they comply to what you have finally decided. This is at the small unit the family level at the social level also there is great degree of compliance and compliance to even absurd practices also. You remember when we were talking about normality we had taken that example of know making the small babies fall from the rooftop of a place of worship. Now this is absurd practice, but you comply to it. Tomorrow we would continue and we would look at know the conformity and compliance factor. We will try to look at know the difference of practice between the individualistic and collectivist culture and if possible we will see two three visuals to make out that in this collectivist society like ours. There could be conformity and compliance even to certain weird type of practices and we do not question it because we show heightened degree of compliance.


Welcome to a new module on linear systems and of first order differential equations and it is a particularly it is qualitative analysis. So far in our all earlier modules the topics we have completed the topics which are basically covered in a university syllabus. So now onwards the two of the main modules are one of the stability analysis basically what we call it qualitative analysis and qualitative analysis of linear system first we will do and then in another module we will learn the qualitative analysis of the non-linear system. So, you can view this module is a precursor to the module on non-linear systems and stability analysis the trajectory behavior around an equilibrium point etcetera. So, that is the main aim of this module it is not just the representation of the solutions which we can still obtain as a solo using the classical theory which we have already completed we can use the existence uniqueness theory of the system of the things which we have studied in the previous module you can apply in a similar fashion here, but the more important aspects about the study of the behavior nature of the equilibrium points behavior of the trajectories because the solutions of the trajectories of ODE systems are solutions. So, you can think it as a motion of certain particles. So, I want to understand that. So, first we will do it in this module about the linear systems and later in another module about the non-linear systems. So, you will also see the power of linear algebra and its diagonalization the Jordan decomposition is usefulness in analyzing these equations. So, what we have done in one of our earlier modules we have studied the first and second order linear systems. So, we can also consider the nth order linear system and what we will see quickly now every nth order linear system is a special case of general nth order system. So, how does a general nth order linear system. So, let us look at the nth order linear system linear equations. So, let me give before that is a linear system a title to this module linear systems and qualitative analysis. So, that is a title of this particular topic. So, how does a general nth order linear equation will look like a regular linear equation will have the following form.  And if you are looking for an initial value problem you can define as you have seen will be seen rather for n equal to 2 for the second order equation not only the initial value problems you can also define the boundary value problem. And you will see few lectures on boundary value problems of the second order equations in a different module. So, for an initial value problem. So, you need n conditions say for example, y at 0 is equal to y naught y 1 y 1 at 0 means y prime at 0 is equal to y 1 etcetera up to y n minus 1 of 0 is equal to say n conditions you need it. So, that is a initial value problem if g t equal to 0 this called a homogeneous g t equal to 0 then it is a homogeneous linear nth order equation homogeneous linear nth order equation. What I am saying is that such a linear nth order equation can be converted into a first order system how do you do that 1 you put x 1 is equal to y 1 x 2 is equal to y 1 not y 1 x 1 is equal to y x 2 is equal to y prime that is nothing but x 2 is equal to y prime x 3 is equal to y second derivative etcetera up to x n minus 1 you will define to be x n minus x n minus 1 to be y n minus 2.  So, if you use this one and we can using that equation here. So, if you look at that x 1 dot is equal to y dot. So, the derivative of x 1 the derivative of x 1 here is nothing but y 1. So, you can write down that way. So, if you look at that one x 1 dot is equal to that d x 1 by d t exactly will be y 1 I am using two notations dot in one both are for the derivatives. So, you can write down this is y 1 that is equal to x 2. So, if you write x 2 dot will be x 3. So, if you go like that you will get x n minus 1 is equal to dot x n minus 1 dot is equal to x n and x n dot has to be recovered from your equation x 1 at that if you look at here this is what your that will be if you take this to the right hand side you can write everything as let me write down that you will get minus p n into x 1 minus p n minus 1 into x 2 if you go like that the last part will be p 1 into x n and then you will also have your g t here. So, this is from the equation this first n minus 1 from the definition the last one will be from the equation. So, if you put this together here and if you define your x t now is equal to x 1 etcetera x n all evaluated at t and you can write down your equation if you combine all these we will write your equation as x dot of t is equal to you can write your matrix a t into your x t you see what is a t where a t is the matrix is a special form of the matrix you have a special 0 1 0 0 then 0 0 1 like that 0 0 the last one here here you will have minus p n minus p n minus 1 etcetera up to p 1 this is correct. So, if you multiply this one a t into x t a t into x t. So, you will have this kind of system. So, that gives you a linear system of equation with a t in this special form see a t is of this one this is your a t you have your a t like that. So, but this allows us to consider any general a t you do not have to consider that way. So, general linear system. So, thus a general system can be written as initial value problem in this form x dot of t is equal to a t x t a t need not be in the above form it can be a a t will consist the entries of a t will consist of functions need not be the form given by the earlier thing x at 0 or you can put it at any point x at 0 is equal to x naught. So, this is a general linear system of course, in general we are not going to concentrate on system such as is this system when a depends on t a plus some g t will also come a non-linear term where you can have some non homogenous this gives you a non homogenous when a t depends on t is called non-autonomous is called non-autonomous. So, if a t is equal to a independent of t independent of t it is called an autonomous system it is called an autonomous system as you see even in our first order and second order equation for example, even when we are studying second order equation when the second order coefficients are independent of the functions you have seen the advantages and able to solve the equations completely. So, there is a larger advantage and many other even the qualitative analysis you see the difference when you study non-linear analysis part of this problem. So, for autonomous system is much more easier to study than non-autonomous system. So, most of these lectures though in this module we will concentrate only on the autonomous systems and we try to represent the solutions and the stability analysis of that system. So, our major aim if you look at case n equal to 1 that means a is equal to a then the solution is you know already then the solution is x t is equal to e power t a into x naught you see. So, our aim immediately we will see that we want to have we want to represent the solution for the system if possible of the form when it is a for an autonomous system. So, our what is our autonomous system autonomous system is x dot of t is equal to a of x t where a is independent of t that is more important thing and you will study the initial problem at any initial value is x naught which is also a vector. So, this system we are always going to refer it to as 1 the important point here is that system 1 is a is independent of t. So, the whole aim is to represent the solution aim represent the solution x t is of the form some exponential form exponential here is what are we have exponential form here what we want to recall the definition of an exponential of a matrix which we have studied already in the basic part of the module. So, if you have a matrix a if a is a matrix then e power a can be defined to be i plus a plus a square by 2 factorial plus etcetera where the series converges that is what a series converges in the set of all linear operators R n to R n. So, in this case given a matrix one can view it as a linear operator from the space R n to the space R n and this is also identified with a set of all n by n matrices that is what we are trying to say in this case. So, on this set of all linear operator from R n to R n we have the corresponding itself vector space which has a topology under that you will talk about the convergence of this infinite series and this e power a. So, note e power a is also a matrix note e power a is also a matrix that is the important part of it. So, you can define e power a in particular we can also define for any t belongs to R e power t a that is nothing, but identity again it is symmetric plus t a plus t square a square by 2 factorial and so on. It is again an infinite series and the series converges uniformly in this case converges uniformly because there is a t parameter.  So, here is an exercise I want to give you an start with an exercise now which you can do it easily define x t is equal to e power t minus t naught of a x naught. Note that e power t minus t naught into a is a matrix. So, that matrix will act on the vector x naught and x t is defined that. So, the exercise is that show that x t is differentiable in t differentiable and x t satisfies one x t satisfies one the system one system one what is a system one this is a system one. So, this is the system one. So, you just compute x dot of t you can see that x dot of t is nothing, but a of x t and definitely x at t naught is easy to see that x at t naught is when t substituted for t naught the first term here will be identity. So, x at t naught will be x naught. So, that way we have the existence. So, this proves this proves this gives existence that is what I am saying. So, you have proved the existence directly by construction of course, as I said the existence and uniqueness can also be proved using the general existence uniqueness theory. But, so the existence which we have already have it. So, and we want to give you a uniqueness a quick uniqueness proof you can get the uniqueness using the general theory, but I want to give you a general easy thing. So, assume y be another solution y equal to y t be another solution. Then what you have to there to prove that you have to prove that y t is equal to x t that is what you have x t is defined already what is x t x t is nothing, but e power t minus t naught of a into x naught that is what you have to prove it. So, proving this is equivalent to that I can take this out e power minus of t minus t naught into a x y t I will take that to the left side is equal to x naught. So, what you want to prove is that this is what you have to prove it. So, this you have to prove it that means left side is a function of t right is just x naught which is a constant. So, you want to show that the left side is a constant and that constant is nothing, but x naught. So, define in that case just the left hand side of z t you define z t is equal to my e power minus t minus t naught of a y t which I can define what I have to prove is that z is a constant in t and that constant is nothing, but x naught. So, to prove that constant here is an exercise again for you these are all simple exercises show that d z by d t is equal to 0 that is an easy exercise when show that that is a d z by d t is equal to 0 then this exercise will give you z is equal to t is a constant, but what is that constant constant, but that constant has to be z at t naught z t is a constant already proved. So, it has to have the same value as t naught, but z at t naught by the definition here put t equal to t naught here. So, you will get this is nothing, but y at t naught, but what is given to you y is a solution to your initial value problem and y at t naught is nothing, but your x naught. So, you have your uniqueness in a quick way without appealing to the general theory. So, we have proved the existence and the uniqueness existence by construction and uniqueness by direct application the linear system the autonomous linear system as a unique solution. So, we may wonder what more to be done here the issue is that show already proved the existence and uniqueness again if you go back to our second order linear equations just obtaining solutions are not enough just like here we have the representation of the solution x t in the form we have the representation of the solution x t in the form e power t minus t naught a x naught you have that one. So, the two of the difficult issues two remarks two difficulties here one difficulty the computation of exponential is very difficult computation though you have a representation but the computation is of exponential of a matrix is difficult exponential of a matrix is difficult in general that is one point. The second point is that as I remarked again x t you have to think it is a dynamical system is a solution to a dynamical system in other words x t you can think it as a motion of a particle. So, the important issue is that how this particles behaves especially near an equilibrium point which we are going to explain to you and you will also learn about the equilibrium points later in the non-linear theory. So, what we are more interested is that the behavior of x t especially near an equilibrium point especially in particular the stability analysis of the trajectory. So, this formula does not reveal anything about x t. So, the second difficulty is that this formula does not do not reveal about the behavior of the trajectories about the behavior of trajectories and equilibrium points which you are going to study trajectories. So, you see that is a whole thing. So, one is especially the computation of this thing and second one is the trajectory behavior and the whole module on this one in the non-linear theory is to understand this trajectory how does it behave what are the equilibrium points how the whether you have the stability near an equilibrium these are the questions from physics engineering point of view. But then there are some matrices where you can have a computation e c example if you want to see. Example suppose a is diagonal suppose a is diagonal that means I write a diagonal matrix in this form suppose it is a diagonal matrix. So, diagonal matrix I will read only the diagonal entries lambda 1 etcetera lambda n this is nothing but what I mean by that is nothing but lambda 1 lambda 2 only along this diagonal lambda n all the other elements are 0.   But then entries with e power lambda 1. So, this is a small exercise which you can immediately do it what you have to do is that you have to compute a square a cube etcetera. If you compute a square you will get the diagonal lambda 1 square lambda 2 square etcetera lambda n square and for any a power n it will be diagonal of lambda 1 power n etcetera lambda n power n. So, and then add it and you get the matrix e power a immediately. But what does this mean when a is what does this mean when a is diagonal as far as our linear system is concerned. If a is diagonal what does that diagonal system means to us diagonal then it just says that x 1 dot is equal to lambda 1 x 1 x 2 dot is equal to lambda 2 x 2. If you rewrite the system n x n dot is equal to lambda n x n u c. So, it is a decoupled system that is no connection between there is no interaction coupling between two variables. Any of the variables do not interact with other variable and each one is a simple equation x 1 dot is equal to lambda 1 x 1 and since x 2 etcetera x n are not coming into picture this immediately this is a decoupled system this is what it says decoupled. So, there is no problem in mainly this decoupling system you will give you the solution x 1 t is equal to e power lambda 1 t into the initial value at the first component that is all. If you go this way you get your x n t is equal to e power lambda n of t the n th component of your initial value. So, if you write this in the form. So, this is equivalent to saying that if you write this e power x t is of course, you have to change here if you if I look my initial conditions at t naught is equal to x naught. So, I have to change accordingly I will change here accordingly this will be e power t minus t naught into lambda 1. So, I will change here this is what I wrote was for the initial value of the origin into lambda n x naught this is just a multiplication. So, if I write the solutions you have your x t is nothing, but your diagonal of e power t minus t naught lambda 1 with n raise this one that is equal to e power t minus t naught of lambda n this is a matrix this is your matrix operated on x naught. So, that is nothing, but your e t minus t naught a into x naught you see. So, you have your solution and you have your complete solution if a is a diagonal matrix and another interesting property with this which will be useful for our analysis we are eventually want to do more analysis. So, if the given matrix is diagonal the corresponding ODE is a decoupled system since it is a decoupled system is solved in each equation each n of the equations independently because it is a first order equation in one variable each one can be solved separately and then you can write down that solution. So, as long as the matrix is diagonal you have no problem of solving it another property which I would like to recall here the property is that in general for exponential map e power a plus b naught equal to e power a into e power b. So, this property of real numbers is not true for general matrices, but if a will be commute a b commute that is a b equal to b a then e power a plus b this will be useful in our analysis later e power a into e power b this you can verify if a and b commute you can prove that result. So, this exponential one of the fundamental property of an exponential function in one variable is not true for exponential functions of the matrix exponentials of the matrices, but if the there is a commutation and that is not contradicting for real numbers this is always true a b is equal to b a for real numbers and that is the another important thing. So, I will recall today some more interesting things which is necessary for here another property which we want to call a property. Suppose b is of the form p inverse of a p such matrices are called similar matrices as we know from linear algebra linear similar matrices represent the same linear transformation it is only according to a different basis you will have different matrices, but it will represent essentially the same linear transformation. Suppose b is equal to p inverse a p then what is e power b I want to know. So, this is a simple result again I will leave it as a small exercise for you if you have not done these things in the linear algebra this may be the correct time to do that one this will be p inverse of e power a p you see. So, you have a nice representation you can do this other way also. So, this is equal to this implies of course, a is equal to p b p inverse and that is also implies e power a. So, you can write down because p is an invertible matrix p e power b p inverse this is one important property which now we are going to use it under this linear transformation this helps. So, you have a linear system x dot equal to a x, but then this allows you to solve for a solving for x dot equal to a x that means namely the system x dot equal to a x can be obtained by solving another system corresponding to b. So, the system one can be solved system one let me recall system one system one means a x equal to x dot equal to a x with x at t naught is equal to x naught this is my system one this is my system one can be solved by solving wide system two namely corresponding to b. So, I introduce a system b equal to y is y at t naught equal to something whatever it is y at t naught is equal to some y naught which we will solve how do you achieve this one. So, recall. So, what is one one gives x dot is equal to a x, but what is x is nothing but p b p inverse p b p inverse of x. So, now put yeah. So, this if you call it here you will get a p inverse of x dot is equal to b p inverse of x not that p is a invertible matrix with constant coefficients. So, p is independent of t. So, you can take inside x. So, you put y is equal to p inverse of x that implies y dot is equal to b y. So, you see and what is y at t naught y at t naught is nothing but p inverse at x at t naught that is x naught and this is your y this you call it to be y naught. So, you can solve your system. So, if you have two matrices are the solution of the system corresponding to one matrix can be obtained by solving the system corresponding to another equivalent. So, such two matrices are called the linearly equivalent. So, this leads to the concept of the leads to the concept of linear equivalence concept of linear equivalence. So, you have a definition for this one two systems the systems one the system one that is x dot equal to a x is said to be linearly equivalent to said to be linearly equivalent to a system to the system two that if there exists an invertible matrix invertible matrix p such that b is equal to p inverse of a p. That is the point. So, whenever two systems are linearly equivalent you can go from one system to another system. What is so important about it? The important thing which you are going to see soon in next lecture or coming lectures one important property the one important thing is that under linear equivalence the nature of the equilibrium points or the stability of the trajectories do not change. For example, if a particular point is equilibrium point there will be a corresponding equilibrium point to the other system and this equilibrium point is stable you will see what are the kind of stability and stability are available soon and then the other point will also be thing and the behavior of how the trajectories will even nature of the behavior will also remain the same. So, the nature the first important property right now you may not know it we as we have not defined any sort of any the concept of stability, but the important point which you are going to see the nature for example, stability or instability of equilibrium points will tell what is equilibrium points equilibrium points do not change do not change under the equilibrium point. Do not change under linear equivalence. So, what is this equilibrium points? So, recall so what is equilibrium points? You will see again as I said more on non-linear systems about it correct definitions again equilibrium points. Equilibrium points are actually solutions of the linear system equilibrium points of x dot equal to A x. So, equilibrium points so though we call it point it is a equilibrium it is a solutions steady state solution. Suppose A x equal to 0 suppose x b such that. So, let me call calculate up not to confuse with initial values x naught b such that A x naught is equal to 0 that means it satisfies the high tensile vanishes at that point. This implies if I define x t is equal to the constant x naught constant will be a solution. That means if you have an equilibrium point and if at t naught if the solution is at that point that the solution will remain at that point all the time it will not move from there. So, that is why it is a steady state solution. So, whenever you have an equilibrium point and the solution starting from there it will not change that. So, for example, for linear system x equal x naught equal to 0 x naught is always an equilibrium point always an equilibrium point. And when A is invertible and this is the case when we are going to consider when A is invertible 0 is the only equilibrium point because if A x naught equal to 0 if A is invertible x naught will be 0. So, if A is invertible 0 is the only equilibrium point. You will see more examples in the next class I want to set a kind of thing therefore, if you start a solution at the origin at time t naught the solution will remain there and here is where your concept of stability and the gravestone. In applications you can never start from a particular point you will always going to make error. So, the question is that if you start a solution at the origin at an equilibrium point then the solution will remain there. Suppose you make an error and the solution is not starting at the origin, but then a solution is starting say at a point initial point x naught which is close to the origin. So, the question is that whether trajectory will deviate from there very far away causing instability because this is important in the point of view of applications you will never be able to start at a particular point you are always going to make error. And this kind of concepts also you have to know that when a solution starting near an equilibrium point will remain there will go to the equilibrium point where it will move away from that leading to various definitions various types of stability when especially when x t is a vector it will have different components certain components will go to the origin certain components will have the stability certain components will not have the stability and all that can happen. And that is what basically in the qualitative analysis we going to see that. The second point so this is about the equilibrium point. So, we are going to initially we will give few examples even when a is not invertible and then you can see the various equilibrium point and how the solution behave. Our more interest is that when a is invertible in that case we have only one equilibrium point near 0 and we are going to study the stability analysis near the point near the equilibrium point. This is the first point second point of interest. The second point is about the change of variable. So, basically we are making a change if you go back to our earlier analysis the previous analysis. See if you go back to this kind of previous transformation. So, you see this is the transformation we made to convert the system x dot equal to a x to a system equal to y. So, there are two linear transformations basically. So, you have the transformation. So, you have transformations x going to y is equal to p x that is one transformation and the other one y is equal to p x that y going to x is equal to p which one is the correct one both are fine y is equal to p inverse of x y equal to p inverse of x does not matter you can replace that x equal to p y are linear transformations are this is important these are linear transformation not some arbitrary transformations why it is linear because a p is a matrix because it is a linear transformation you can view this as a that is a important one can view it as a coordinate change view it as a coordinate change. So, basically when you are considering coordinate change. So, when you have a some coordinates you will be basically looking at two coordinates change either x going to y or y going to x. So, you can see that if this is y 1 y 2 you will have a coordinate change x 1 x 2. So, effectively when we are considering under linear equivalence x dot equal to a x and y dot equal to p y we are making a coordinate change under this coordinate change the when you for example, if I is invertible and x naught is an equilibrium point and you will have a y naught is equal to p inverse of x naught will be an equilibrium point that are the only equilibrium point you can get that equilibrium point in the stability system x naught is stable which we will see y naught will also be stable that is what I say it does not change there may be a difference in movement of the trajectory for the stability analysis you cannot convert a stable equilibrium point to an unstable equilibrium point under linear equivalence. So, the fundamentally is a is a coordinate change that is what under this linear transformation one has to understand and that is not surprising because this linear equivalence is between two similar matrices as you see that the similar matrices represent the same linear transformations hence you should be able to do the same thing. So, what is our aim now with these are general remarks about these are basically what is our aim we should not forget about aim what are we trying to do this one see what we have seen is that you are interested in solving a linear system x dot equal to x and that you have a representation you have a solution x at t naught is equal to say x naught you have a linear system and what we have so far discussed is that if there is a linear equivalence linear equivalence that means you have a b is equal to p inverse of a p and you have the corresponding system y dot equal to b y and y at t naught is equal to some p inverse of x naught that is your y dot. And here the solution as I said that the one of the major difficulty is the computation of the solution even though you have a solution x t is equal to let me recall again once again because this is the important point x naught is equal to a into x naught the computation of this is difficult suppose you are able to transform this under linear equivalence you look for a linear transformation. So, that the computation of e power t minus t naught into b into y naught this is called y naught. So, this may be easy this may be difficult may be difficult probably easy you see this is a situation we are looking at it. So, the question is that is it possible to find a linear equivalence. So, that the solution to y power dot is equal to b y is easy to compute when I say that solution to y dot equal to b y is easy to compute this is easily computable. And we have seen one situation where this can be computed easily when b is diagonalizable. So, this leads the question of diagonalizability of a matrix. So, that is where we are looking at it. So, that is where we are looking. So, the so far we are discussed this is a linear equivalence. So, that we are looking for a matrix b corresponding to a using the linear equivalence. So, that b is diagonal you see suppose b is diagonal that is a question suppose b is diagonal lambda 1 etcetera lambda a. So, then what is solution of y t the solution y t is immediately you can write it y t is equal to the diagonal of that one. So, diagonal e power t minus t naught lambda 1 etcetera e power t minus t naught lambda n of this is a matrix acting at y naught what is y naught p inverse of x naught. So, this is what your y t and what is your x t x t is nothing, but p of y t that is what our transformation. So, you have your solution. So, you have your solution immediately p diagonal e power t minus t naught lambda 1 etcetera e power t minus t naught lambda n of p inverse of x naught. So, you see you have your complete solution here you can write down if you can question. So, that is why this is the question of diagonalizability immediately. So, the diagonalizability the importance of you would have seen the diagonalizability importance not only here when you are studying the linear algebra whether if you can solve your linear system if the matrix is diagonal because there again is a decoupling system exactly what you are showing that. So, the question is that given a matrix given a matrix a thus there exist an invertible matrix invertible matrix p such that b is equal to p inverse of a p is diagonal. If so our problem is solved our computational problem is solved because now the computation of e power b is easy because b is diagonal, but unfortunately in general the matrix need not be diagonal the matrix need not be diagonal. Example even when n equal to 2 that is the diagonal need not be diagonalizable need not be diagonalizable that is what I said not diagonal when n equal to 2 if such a matrix exist is called diagonalizable. Example you take lambda 1 0 lambda is not diagonalizable how do you prove that what is the kind of diagonalizable. You see even such a matrix simple simple looking matrix simple looking matrix is not diagonalizable. So, again let me recall from linear algebra diagonalizability is equivalent to the existence of n independent Eigen vectors. So, remark so recall diagonalizability is equivalent to the equivalent to the diagonalizability is to the existence of n independent Eigen vectors. This you are already seen in linear algebra we also recalled some of these things in our basics independent Eigen vectors. So, here is an exercise for you for the problem for the above matrix for this matrix for show that it has only one Eigen vector show that it has only one Eigen vector one independent Eigen vector independent Eigen vector you see that. So, when you are studying the linear system our main aim is that given a matrix can we have a diagonalizability property. The moment your given matrix is diagonalizable that means you can find a matrix B. So, that is a corresponding B you are completely the system is solved if the system is not diagonalizable what can we do about it. This is where the linear algebra and the Jordan decomposition comes into help and we will use that to understand the how much we can do it if the matrix is not diagonalizable. What we will do in the next lecture we will do a complete analysis in a 2 by 2 system because 2 by 2 system the possibility what are all the possibilities is easy to classify. Even in higher dimension the Jordan decomposition tells you the all the best possible way you can do it and but we will spend a more detailed study in 2 by 2 systems and in the process we introduce the concept of phase plane and phase portrait and then the various stability. So, we will have a complete analysis in the next lecture. Thank you.