Good day. I'm Colonel Jerry Morlock, the Director of the Combat Studies Institute. You're about to use a video series which our instructors have prepared for the sole purpose of improving your presentation of M610, the evolution of modern warfare. We've taken care to make the course that you teach as similar to the one taught at Fort Leavenworth as possible and choose to add these tapes to your libraries in order to give you every advantage as you prepare to teach this new course. These tapes are similar to the weekly train-up sessions which we utilize to prepare our instructors here at Fort Leavenworth. My intent for the tape sessions was to provide you insights and tips on ways to approach the lessons of M610 that were not available in the instructor notes. I've drawn various instructors, military and civilian into the sessions based upon their specific expertise and historical background. They were asked to just talk to the lesson structure and content, giving you some additional information on the historical context and differing views on how to approach the lessons. These tapes will provide you a wealth of knowledge and direction that will significantly improve your readiness to teach our new history course. One word of caution regarding how to use these training tapes. They are not designed to be substituted for your instruction during the individual lessons of the course. As instructor preparation tapes train the training material, if you will, they are inappropriate for direct instruction to students and are not intended for that purpose. Our intent with these tapes is to improve your ability to lead the student seminars by sharing tips and advice from some highly qualified experts. The Combat Studies Institute stands ready to provide whatever additional expertise or assistance that you may require, and we've included the institute's phone, mail, and email contact information on the tape if you should need it. Good luck with the evolution of modern warfare course. Have a good time. Greetings and welcome to Lesson 3, The French Revolution and Napoleon. My name is Dr. George Gavrich. Joining me will be Lieutenant Colonel Scott Stevenson. If you remember, we were with you with Lesson 1. Based on that performance, I know they can hardly wait for what we have. Exactly. Since last time we talked about limited war, it seems like we shouldn't stick on the same message, but talk about something different. What we're talking about is a revolution in military affairs of a tremendous, tremendous scope. If you remember when we talked about the age of Frederick the Great, there's an age of limited war where armies fought often for just the territory next to the kingdom. The battles basically lasted less than a day. Most of the battles were sieges. On the rarer side, did you have open field battles? With the French Revolution and the emergence of Napoleon, the warfare is going to take on a whole different dimension. We're going to be talking about Napoleon being able to raise a half a million men army plus and march all the way to Moscow, while at the same time having a half a million army down in Spain. Something unheard of. A war economy that can sustain battles for several days. Napoleon said I could lose 30,000 people in one day and fight again another day. Something Frederick could not afford to say. Something monarchs before him could not say. So we're talking about a transformation in warfare from an age of limited war to what some people say starts to move toward total war, general war, where battles and armies are on a whole different scale than ever before. The way I think this lesson could be organized is it's important to see that any great commander like Napoleon, a genius, military genius is a product of his time. There's a little bit about the French military and the innovations they're trying to bring in then talk about the French Revolution. Some say they couldn't have been known to Napoleon without the French Revolution. What impact does the French Revolution have on warfare? So in the first case, we're talking about some military influences. In the second case, non-military influences as they shape and help explain this new warfare that emerges. And then we need to talk about Napoleon himself and the system that he creates. What he bring to make for warfare on a whole different scale and then end by looking at how he gets defeated at Waterloo and his undoing. Let's look at first the French military. If you remember Napoleon, not Napoleon, I should say Frederick the Great beat up on the French military. After it was all over with a good number of French officers serious about their profession, thought about how they could reform so that they would not be beaten by Frederick again. I think it's important to talk about some of the changes that are brought about in the French military because Napoleon is going to be raised as a young officer in this monarchical military, the Ansonne regime, and he'll take ideas of those reformers and put them together. So let's talk a little bit about the French military before Napoleon and the French Revolution. There's a couple of figures that get mentioned or several figures that get mentioned over and over again. One of them is Gébert, who is making some stabs at the military theory and we talked about military theory. Could he be like the Tofflers of back then or great thinkers looking at the future and the present and making sense of it? Depends on how you look at the Tofflers, doesn't it? Okay. All right. But here's a fellow who's thinking about the future. War says that war may take on a whole new scope that once you get the people involved. And in Frederick's time, normally you try to keep the people out of the battlefield. You have trained professional armies. And again, the ideal is to be able to march an army through a province and have the population not even know about it. But what happens if you get the population involved? This may throw things off. Gébert has got some theorizing about that. Of course, he can look across the ocean in the last lesson, lesson two about the American Revolution and see there is a people involved. And what does that mean for the future warfare? Absolutely. There's another fellow named Griebeval who institutes a whole bunch of changes, especially in French artillery, toward making it more mobile, more of an instrument of supporting maneuver on the battlefield. He lightens up the caissons. He lightens up and standardizes the different pieces of artillery and makes that a much more useful, effective tool than the heavy guns that had to be dragged around the battlefield. And once you set them in place, basically they standard for the whole battle. Under Griebeval, French artillery is going to probably be the most technically advanced in Europe. I think the interesting point there is Frederick was very good with his infantry. They were the centerpiece of his army. The French cannot drill to that level, they feel. Is there a cultural problem? A cultural problem. So they can fall back on what makes their strengths. They got famous engineers and they got famous artillery people. They've got more resources so they could build more artillery pieces. So what they end up doing is plant to some of their strengths, building up the artillery. They have the resources to have more artillery pieces, standardize it, make it lightweight. And you've got some artillery theorists who are saying, if we can do these things with artillery and start massing it because it's lighter, faster, we could maybe make artillery the decisive arm on the battlefield, a kind of a revolutionary thought. Napoleon is what an artillery man. Indeed. He is exposed to these ideas, he's exposed to these reforms and he is one of those experimental units that's playing around with these artillery pieces. It's also interesting, he has a Corsican, a guy who comes up, shows up at military school with a heavy Italian accent. He's sort of an outsider and he doesn't have access coming from minor Corsican nobility to the more fashionable regiments, either infantry or cavalry. So for him, going into artillery, which accepts middle class and lower nobility is kind of a logical thing. But he starts out as an outsider. Here's a guy we see early on with kind of a fire burning into him and he's willing to accept these changes that he sees going on in the French Army and he's doing some thinking of his own. He's doing a lot of intensive study of military history in particular. I don't emphasize that fact. Like Frederick the Great, we should emphasize that too. We should emphasize that as well. I think we're on to something. All right, but as you say, he comes up under the old monarchical army but here's a guy who's already thinking, perhaps beginning to think outside the box, but it takes a series of circumstances that sort of give this young Corsican upstart his opportunity. And here's where you're, as an instructor, you've got some decisions to make. You may want to fill in the blanks for students about what the French Revolution is about if you feel up to that. Otherwise, I'd get past it as soon as you can. You've got to mention it at least in passing because it sets the stage for what some people consider to be the greatest general of all time, Napoleon. What some people refer to as the God of War, Napoleon Bonaparte. He's a product of his upbringing in the old armies, a product also of the revolution. The French Revolution is a complicated thing and a whole library has been devoted to it. A lot of it has to do with the financial crisis that hits France in the late 1780s. Part of it brought on by supporting our own revolution, part of it by a long series of wars attempting to gain some domination in Western Europe. Financial crisis that allows some classes in French society that haven't had a voice, middle class and lower nobility, rise up and attempt to seize a larger measure of power. And it's also, I think it's an agrarian on an economic crisis, lead to a political crisis. The king attempts to flee the country. Louis XVI, he gets captured, he gets beheaded, and all of a sudden all the crowned heads of Europe are starting to get a little nervous about what does this mean? When people can rise up and overthrow and then execute a king, the whole system is at risk here. So all the major powers unite to put down this French Revolution. Sir, like the Bolshevik Revolution brings in the rest of the world against it, and the Iranian Revolution. When it emerges, people want to stop it. You don't play with revolution. This is a key. And then here. Again, we talked about Frederick VIII in the Seven Years' War. You look at the correlation of forces against him, he doesn't have a chance. You look at the same thing with France in 1793-1794. Its army has not fallen apart, the old monarchical army. It's lost its officer corps, about a third of its enlisted ranks have gone off and deserted. It is in bad shape, and all of a sudden the various houses of Europe are marching on its frontiers with big armies, and it looks like this revolution is going to be stamped out just as quickly as it rose up. The question is, how does a revolution like this made by the people, how does it defend itself? What do you think? Well, I think one of the things that is important to ask is, what does the French Revolution do to the character of war? And there is, first of all, a major change in the political system. We see nationalism rising up, the power of the people. This is really, you could say, the emergence of the nation state. What then that means in this revolutionary fervor, is people see, wow, the old monarchies going away, a lot of the aristocrats are fleeing. I have a stake in this revolution. People who are nobodies are all of a sudden head of revolutionary committees. They have a stake in saving this revolution, because this is a government for the people. So what you see is where the peasant tolerated soldiers marching by, or I should say probably more in the cities, tolerated soldiers, now he's willing to take up arms and join and go out on the field and fight under this elan, the spirit of revolution and nationalism. And it doesn't need to be motivated. It's got the motivation within. Now you've got to somehow drill and train them a little bit, and you've got enough of the old guard to help with the professional side, but you've got a lot of now these guys willing to fight. So there's a zeal that takes them for a few years and helps them to win on the battlefield. That's one thing. So the nature of the political state has changed, which means if you want to fight Napoleon or before the revolutionary regime, a lot of military people who are progressive will say, you've got to change the way we do business with our own people so we get them on our side and we see this flow of the nation state into the rest of Europe. That's one thing I think. And then the second thing is if the state has changed and become a more nation state, the army has become national. You no longer will need mercenaries as time goes on because you will conscript people from within.  They naturally rise up when the call comes and says we need to fight for the revolution. They'll start going out. A coercion helps sometimes. A little bit, a little bit. But as time goes on, what has happened? You haven't changed the industrial base. The agrarian society has not really changed, but what has happened is people have been fired up within a short period of time and now we have a new standard, a new mindset.  What then happens is you can now take that fervor. You could take that mass army that's a million strong armed and now put in the mechanisms, the coercion, the conscription. So from now on you can mobilize the society and raise armies on a much larger scale, which means you could take losses on a much larger scale than before and replenish them with conscription. So we have the emergence of national armies and wars of nations. Whereas in Frederick's time, a single battle can decide an entire war when the French Revolutionary Regime can put in the mid-1790s a million men into the field. All of a sudden, you can't beat those guys in a single battle. You're going to have to organize a campaign. This is, you've got a whole new order of problem here that you've got to take on when you're facing a million.  Let's face it. How are you going to take, you've got, you've built an army, but if it's still these untrained citizens, how are you going to turn this into effective force? And that's the problem that the Revolutionary Generals faced. It's a thorny problem, a pressing problem, especially because the Revolutionary Leaders tend to take Generals who lose in battle and they tend to make an appointment with Madame Guillotine.  One of the solutions they come up with, it's relatively ingenious, is they'll take a battalion of the old, regular French royal troops and they'll put them with a battalion or two of these new conscripts to create what's called a demi-brigade. The idea being that the well-drilled, well-disciplined old regime troops will serve as an example for these new conscripts who, as willing as they may be, don't know a thing about fighting and make them their example of these royal troops, these old royal troops, will turn this demi-brigade into effective fighting force. So what you're facing, if you're trying to take down the French Revolution, you're facing huge armies, you're facing armies that are motivated by something you haven't seen on the battlefield, patriotic fervor and nationalism, and the nature of these troops allows them to be used in ways you haven't seen before, either. I think the interesting thing, too, is if you compare it to the Bolshevik Revolution or the Iranian Revolution, you have their emergence of people outside of the military in power who don't trust the military, political commissars, religious commissars in there. The French Revolution brings to power Napoleon, who's a general, already a proven general. He'll take the talent that's there that's demonstrated by these young officers who fought and won on the battlefield, have been elected, promoted, and he'll use their talents to mobilize them into a better army, a better trained army, and use that fighting power now to conquer. Rather than fear the army, subdue it, he will use all its resources to the maximum, which is interesting. Napoleon is probably a guy in the old royal army probably doesn't get beyond captain or major in the career. All of a sudden, though, due to being at the right place at the right time, defending the national government against the coup attempt, with the famous Wiffel Grape Shot, all of a sudden gets elevated to general, gets a theater command down in Italy, becomes a national hero, and this is a springboard eventually to take power. But he's going to bring his own genius to this piece, but you've got to remember that he is inheriting, like Frederick inherited the system his father built, Napoleon will inherit some important advantages from the French Revolution. Number one, they've answered the problem, where does the manpower come from? Number two, where do the Oscars come from? Well, as George has said, you now recruit across all classes of society, use some of the old Oscars if they're willing to serve, most of them have gone overseas or been beheaded, but you recruit by talent now, by merit, which is a revolution in itself. Okay, I got the people, I got the Oscars, how do I arm them? You orient your entire national economy towards supporting these huge new armies you're putting in the field. This is the first appearance of what some people call war socialism. A war economy. And that's done by a guy named Carnot who's called the organizer of the revolution even before Napoleon takes over. He is now helping to organize the state to keep these guys funded. Okay, so I got, well, funding is another piece too, but we're going to orient our government toward producing resources. So I've got the manpower, I've got the Oscars, I'm going to orient my economy toward keeping them well equipped. Now I've got yet another thorny problem, here again there's another problem. And this sort of has the impact on strategy. If I got this huge mashup of guys to feed, I'm going to be benefited by the fact that agriculture is taken off during this period of history. It's going to be easier to feed the guys if I let them loose across the land. Now, further it couldn't do that, but I can let these guys go out and forge across the land and send them out, hey, Pierre, go out and find us some chickens for tonight. And Pierre, I can have some reasonable expectation. Pierre, are you going to show up later on tonight? Later on tonight. Well, Pierre being a resourceful guy that he is, I would hope so. By the same token, the road networks have increased so this increases my ability to disperse armies to go out there and forge. But still, you know, the trouble is a forging army is going to leave a barren countryside behind it. So where do I want to deploy this army? Well, the ideal place is not in France. I want to take this, I want to take my act on the road, you know. And if I'm going to have this huge army, I'd much rather fight with it on foreign soil. Let, you know, let him do his forging in Germany or Poland or Italy or anywhere, but in France. Now, if I have an army that relies on forging, I still have a problem, because if Pierre is in the lead brigade life is good, you're going to find plenty to eat as you get on the road. But if you're Gaston in the trail brigade, suckin' dust, there's not going to be much left when you get there. That's right. What's the solution? Well, this is a thorny one too. A lot of starving goes on in these ill-managed armies. One solution is start putting your armies on multiple axes. It's just, it's a fact of life. If you want to feed them, you can't put the army on a single route like Frederick used to do. Frederick could rely on this carefully organized system of military depot. Well, a French revolutionary army can't do that. So by necessity, they're going to use multiple routes, but Napoleon's going to turn that into a benefit, isn't he? Yes, he's going to create the core system. And what that means is you can create independent armies that march down different avenues to congregate, to mass at the decisive point, or for the engagement. What's interesting then is you've got to make these cores be able to be fighting independently. Well, why is that, George? What happens if they get caught by themselves out there in the middle of the night? Get the enemy army. They get defeated piecemeal. Well, yeah. If it's just a force of infantry, the enemy using combined arms will overwhelm with artillery and cavalry. Right. So what you end up doing is making these cores combined arms.  You've got basically independent command, a core system of combined arms. Whereas, go back remember when we were in lesson one, was it with Raspach? Well, when you looked at Raspach, you had what? On the hill was the artillery, seedlets in his cavalry were hidden behind a hill, and the infantry was in a different place. All the arms were there, but they were not integrated at the level of integration that we're going to see in this army. It means now, independent command, armies not marching as one single army, but a multiple axis, each core independent. I still got a problem with that, George, though. That's why you... If I'm Frederick, I know I've got everybody in sight. I can keep track of my army. I can orchestrate its efforts on the battlefield. But if I let these guys go marching off across country under separate commands, how do I keep control of this mess? First of all, you've got to get good intelligence. You've got to have a good career system, and you teach them that you march to the sound of gunfire. So, what you'd have to do is find subordinate commanders who can use their own initiative. And, well, you don't have to necessarily find them. They've all in a way been developed by this long series of revolution. This service for you, see, you've got your future marshals there. All you got to do is just kind of give them a promotion, a marshals baton, and they've got the experience to be able to fight independently. And then when the fighting starts, you teach them you march to the sound of the gun. So, that's kind of like a satellite message coming down to the commander. There's fighting over there. I know it's fighting. Napoleon wants me to move there, so I move over there. Okay. Well, that's good. What does that mean then? It means it's going to be more difficult when that enemy comes out as a single army. If you look at that Prussian army coming out, it's in Frederickan spirit. It's well-disciplined, finely tuned, old generals who believe in Frederick. You know, this is the army of the 18th century. They're going out to meet the new army and they can't figure out because it's going to be coming on multiple axes. Where's the main? That's the example I use in my students. If you're looking for a Frederickan army, it comes at you like this. Okay. It comes at you like this. Is there any doubt where the main effort is? Now, it's right there. In that fist. It goes like this. But what you see with Napoleon is that you see multiple columns. Okay. So I'm going to commit where's Napoleon's main effort? Is it here? Is it here? Is it here? I'm not sure. If I attack this guy and I guess wrong where the main effort is, the rest of the guys will collapse on me. That's part of Napoleon's part of his part of his system. He moved by cores that would move in a box, say a core at each corner of this book here. Let me put it down here where you can see it. And if you go after this core right here, the other ones are trained to move against your flanks. Wherever you hit it. If you hit it here, for example, the other guys will move to the sound of the guns. They're about a day or two within each other. Each core is designed to fight independently. It's designed to fight just long enough for the rest of the cores in that box to come to his rescue. What happens if I'm coming at you in a coalition? Here's a Prussian army and here's maybe a Russian army. What does the core system allow Napoleon to do? A good question. And you'll see this again and again. He's more powerful. You see Napoleon using this in his first campaign in Italy. You see Napoleon using his last campaign at Waterloo. It's fix one force or slow it down and mass on the other force. He is a master exploiting the interior lines exploiting the enemy, the division between enemy troops and coalitions, and Napoleon turning around and beating the other. He'll do it again and again and again and again. We've talked about his system. I think we ought to talk about the guy himself too. What makes him special? We said he inherited all this good stuff from the French Revolution. He's just a lucky guy. We happen to be at the right place at the right time. Or does he bring something special to the art of war to the battlefield? What do you think? I think first of all I would say he's partly a product of his time. And that's not to demean his accomplishments. We're all products of our time for better or for worse. So he does benefit from the French Revolution and what it brings to warfare. He does inherit some good ideas and reforms from the French military. So I think yes you have to look at he is partly a product of his time and he's partly lucky. But there's nothing wrong. What is it Napoleon once said if I have a choice between a good general and a lucky one I'll take the lucky one any day, any day. Now I would say that one thing too they say about Napoleon is that through the study of history reading a great deal studying the masters including Frederick he developed in his own mind how he was going to command. And then once he starts to command and lead he uses those experiences to fine tune. And in a way that's the same scenario you see in a way with Frederick the Great. He got the tutors he studied the history he went on campaigns with Eugene to observe the first battle comes he learns a lot from it but he's got the base of knowledge he's been tutored educated well in Napoleon's case it's more self-taught so that's one I think the thing you have to keep in mind and that's the importance of study that's why all these great guys who are successful try to impart to their subordinates a study. What else do you think makes him I keep a vivid picture in my mind of Napoleon preparing a campaign when he anticipates he's going to go to war say against Austria or Russia or Spain what he'll first do is assemble all the books he can on that area the geography the topography the social organization then he assembles all the books that can be found on previous campaigns have been fought there. Campaigns in Frederick's time, campaigns in ancient times then he assembles the best available maps and he will spend apparently according to some accounts day after day on his hands and knees pouring over these maps figuring out the distance from point to point pouring over these accounts what's the terrain like what are the key points where are the river crossings where's the enemy likely to deploy and as he's doing that he's starting to send out his cavalry and his agents to go out and start developing intelligence he is a one man staff now he has a chief of staff guy named Berthier who's very good at transcribing his orders but what this guy brings to battlefield is a sense of time and space that normal people don't understand an understanding of if I move this guy here, move this guy here where's the enemy likely to be how fast can I react and then move over here he keeps it all up in here something that mortal men at this time have real difficulty doing but he's developed that ability by spending all this time studying the maps developing the branches and sequels the campaign as he goes along and it's amazing now he's got hundreds of guys working for him whether they're couriers or scribes or what have you but he's keeping the essential pieces up here and he doesn't have a staff the way we understand it today when it comes time to develop courses of action he asks one guy himself what are the courses of action and he starts working that as it is in a way you could almost say that the modern staff was built to try to simulate what Napoleon did by himself and the depressions for example we'll talk about this in later lessons have to come up with a staff of specialized decision maker people that support decision making as an answer to this one supreme genius that exists on the other side because Napoleon again, he is the master of the time and he brings enormous energy here's a guy in his early days who could spend all day on horseback and then all night dictating orders enormous physical vitality enormous vision enormous charisma here's a guy who knew how to motivate men I think that's a good point to talk about how he motivated troops they said he had an amazing charisma a magnetic personality met him in court, you were dazed by him on the field he could ride up and give a Marshall's baton to a soldier for doing heroic things and all of a sudden the word spreads to someone for great valor on the battlefield he's the one who brings in ribbons to decorate soldiers he's able to reach out and touch the individual soldier and motivate him because you know motivation is difficult and that fervor of the revolution died within a couple years of the revolution and a lot of these veterans at the end they are fighting for the emperor there were veterans who go to battle because it's him one of them was sent out to stop Napoleon as he was marching back into power to capture him for the king and he just kind of crumbles in front of him and turns over and says I'm going to be on your side now so his ability to touch out and motivate the individual soldier was a gift and it was a highly motivated army when it was at its peak the other thing I think that was interesting is that he had almost a photographic memory in detail just like Napoleon just like Frederick went down to the battalions watched him march he understood his entire army Napoleon could remember details and he'd go through logistics to walk through a regiment and remember the names of the veterans and then there's a battle going to take place potentially a battle at some seaport he'll ask about are those batteries still on top of that hill so here's the thing that as you go up higher in rank there's another piece to it too and it's in your instructor notes that I'd add as kind of a counterpoint to what you say his capacity of detail is famous quote by him he says there are in Europe many good generals but they see too many things at once I see only one thing namely the enemy's main body I try to crush it confident that secondary matters then settle himself for all this capacity of detail here's a guy who could focus in on what's truly important and he could let all the little worries about this decide hey this is what I need to do and I'm going to take action the French have a word with court kind of intuition about what reports to discount what reports to accept what are the key matters of this fight here there may be some fog and friction out there but I can focus in on what's really essential and going after the enemy army there's one real change he's made to warfare in Frederick's time if you want to avoid battle because an army was a big investment so you tended to fight sieges that ended up deciding the fate of a local county or a fortress or a province in Napoleon's time in Napoleon's way of war there was one object it's the enemy army I crushed that then I put my foot on the enemy's neck and I dictate peace I hit him with a decisive battle crushes army then I can have my way with him I have the troops to do it now the French Revolution has given me this huge army I can risk battle now I can go out there and go for the decisive action and I'm going to hit him with this thunderous fight stun him into submission dictate peace to him and he'll do that again and again for the successful years of his reign he'll go out there and fight the decisive battle his masterpiece is Austerlitz which is in your read I think one of the things that to me is interesting is he's trained a different army his speed doesn't come like Frederick's speed in marching with a quick pace and a disciplined fire of his infantry he has brought speed with the ability to bring combined arms bring course together onto the battlefield hitting you from all sides at the operational level hitting you with all those combined arms in an integrated fashion on the battlefield and with artillery being so much more important you don't have to spend as much time training your infantry like Frederick did you make it more of a light infantry all-purpose infantry so you don't have to train it as much it can be replaced much faster than Frederick's so now if you want to fight against Napoleon you've got to mobilize your society differently you've got to be able to sustain losses at a much higher rate and you've got to be able to replenish and adapt some of the things that Napoleon does which means you take a little bit of the course system you take his infantry artillery and our cavalry tactics incorporate them yeah and if you're a European monarch trying to fight this is a brand new way of war and it's stunning it's faster than what you've seen before it's more massive what you've seen before the tempo of warfare is like you've never seen Napoleon's constantly operating inside the decision cycle of his enemies who are used to moving in that methodical single column from siege to siege now here's a guy who comes at you in multiple columns and wants to fight the big battle of what you're going to do the case study that you're offered Napoleon's masterpiece and I offered it's an amazing example of how Napoleon sees a battlefield, has pictures in his mind, establishes what we call in modern parlance information dominance has an outstanding deception campaign and then on the tactical battlefield we've talked about his operational field for a fight and when to bring the course together but the tactical field the minute to minute awareness how long his units are how long his units can hold out when the enemy is going to be in a really vulnerable position and then deciding at the exact right moment to commit his reserve into the enemy weak spot to exploit the situation he's already set up days before the battle even goes on he also is aware that the enemy's decision cycle because it's coalition, it's Russians and Austrians it's the old style command and control where the monarchs, the Austrian emperor and the Russian emperor are at least in name if not in fact in charge there's no coalition doctrine this guy could work off of fails to help integrate them I think the thing that I think is important when I look at coaching if you've got a talented team like the Bulls just won there's so much more freedom that a coach has in how he plays the game he's got the confidence I think what you see in Austro-Litz is a confident Napoleon who can take a risk it's amazing we say in warfare you go for the decisive point you take the high ground Napoleon was sitting on the high ground he's afraid that the coalition won't attack him he wants it to attack him so he comes off the high ground and lets the opponent take it which already now becomes seductive to the coalition Napoleon must be weak he's withdrawing from the high ground then he starts digging trenches like he's preparing for an attack on him rather than preparing himself for the attack and that's amazing and that takes risk and only a person who has confidence in himself and in his system because I think as we'll see in Waterloo it's a tired Napoleon but it's not the same army he doesn't have the same generals under him he can't take those kinds of risks so units that are well trained well motivated give a commander so much more opportunity to take risks because he can rely on those people to do their thing and in each of these cases like in Rossbach, in Leuton here we'll see in Austro-Litz these subordinate commanders if you got into the detail of the battle are helping to bring victory to the supreme commander Sideless does it at Rossbach you've got a couple brigade corps commander that show initiative and heroism to help pull this thing off for Napoleon but it still reflects greatly on the man himself who crafted this army this is his grand army he's taken quite a bit of time to shape it, train it, mold it to be his fighting machine well if it's so wonderful how did anybody ever bring the guy down do you have to be as good as the Germans were in 1940 of course as good as the Bulls were this year to beat him in 45 do the Allies have to be as good as Napoleon was in 1814 to defeat him in 1815 or as good as he was in 1807 to beat him in 1815 right and I would say no what you find is on the one hand there's going to be some decline in Napoleon and his system he will never have an army as good as the one he took to the Austro-Litz why not, why not not to the level they were in 1805 the army that he takes to the Austro-Litz is an army that has spent a period over a year it cites along the British Channel going from brigade to division to corps size maneuver it's an army that's intensively drilled it's an army that is made up of in large part of veterans from the previous wars of the revolution it's an army that's got officers who have been brought up through that talent search that Napoleon and the previous leaders have done great leaders, he's got well trained troops he's got the system in place he'll never have an army that good again you could create it but there is a problem I think in that the revolution created the talent he did not create a system to keep that talent coming up there's no staff college there's no great educational system so what that means it all depends on Napoleon and when he's losing or there's marshals or they're starting to age it's good he does not have the ability to create a new generation of leaders to replace them he is too much dependent on himself he's probably too vain too arrogant to learn from his opponents to give them credit plus to be able to recognize talent beneath them so that's going to be one problem you're not going to have the people underneath you Frederick the same he dies basically with a pretty good army you don't see any great leaders emerging after Frederick the guys that are in charge are old guys who just want to imitate Frederick they don't want to change the system which in large part explains why the Prussian army gets spanked so bad in 1806 gets a national humiliation at the hands of Napoleon that's the second of his two classic victories first Austerlitz in 1805 then there's Jaina in 1806 and he's taken on the best armies of Europe he humbled them he's the master well, again just like Frederick had to face in Austria that's constantly getting better, raising the standard Napoleon is going to have to face enemy that are forced if they're going to face him to improve themselves Austria is one example, Prussia is another and there's Napoleon himself his eyes get bigger than what he can digest he looks around, he says I'm going to make Europe one gigantic embargo against the English who the only guys at that point were holding out to him by 1807, 1808 people holding out against Napoleon are the dead British, as they say he can't get at them because he's thwarted by Nelson Trafalgar he can't build the French navy he can face him so he's going to put a blockade on it he figures he will humble this nation as shopkeepers with the continental system probably the continental system there's some leaks in it the most prominent at each end of Europe one end there's Spain and Portugal and at the other end there's Russia I got to bring these guys in the system in 1808 he goes into to go through Spain to get to Portugal to clamp down on them ends up in a campaign that turns into a peoples war the Spanish people rise up against him after he deposes their king he gets into a long bloody brutally ferocious lots of atrocities in both sides guerrilla war in Spain that eats up by some estimates as many as half a million French troops over the period from 1808 to 18012 the Spanish ulcer that eats away at his army at the other end is Russia that's the whole different thing isn't it what's interesting there is with the vastness of space there in Russia Napoleon just gets dragged deeper and deeper into Russia not necessarily by design until he extends his line so far that he can't get back it's reminiscent of what we'll see in World War II with the Germans losing face in 90% of his army he goes into Russia with the same plan he always does in a campaign I'm going to go in there with a massive force and force the enemy into a battle but the Russians say, well we're going to fight you but I think a little further back and they get ready to fight a battle the Russians don't have this plan from the start of a scorched earth back to Moscow it works out that way though Napoleon keeps having this idea I'm going to bag the Russian army keep slipping away from him he's all the way into Moscow the question is what now G.I well if I take the enemy capital he'll be forced to make peace the Russian Tsar says, not necessarily and here's this great genius and he has to scratch his head in Moscow trying to figure out what the heck happened I won all these battles I'm sitting in the capital and he's still out there with his army and he wants to fight and it's getting cold I better retreat back and all of a sudden he finds it's a different ball game yeah and he's in a large extent he's a victim of the kind of forces that helped him along this nationalism nationalism in Spain has given him fits that's right and nationalism in Russia has given him fits here's the people, it refuses to give up even after their armies are beaten in the field and their capital is taken the Russians refuse to throw in the towel and then as he falls back with this beaten army out of Russia the Russians say, wait a second we're going to rise up in a spirit of nationalism and the Austrians do the same thing and all of a sudden by 1813 all Europe was ablaze against him and he's lost the biggest army he's ever put in the field he's put some estimates between 500 600,000 men into Russia lost virtually all of them he's got to rebuild an army from scratch well he does that he puts 400,000 men into Germany he fights a huge campaign in Eastern Germany it culminates in 1813 the battle of Leipzig which is the biggest battle in history to that point with about a quarter million people on each side he loses falls back into France, fights another massful campaign but at this point the French people are running out of gas the nationalism, the fervor that he'd relied on the desire for glory is just about gone at this point well it'll flicker up one more time he gets kicked out in 1814 he'll come back in 1815 that sets the stage for the Waterloo campaign and we get to see Napoleon at his best and his worst too I think I think that's interesting too is that manpower there's a limit to how much he can fight, he can stay fired up with war reap the benefits of war and see its people being killed off in war so you find the character of his army changes he starts bringing in what you call the mercenaries forcing Poles and others to fight so it's no longer the national army of the French Revolution the era period of Napoleon then I think if you look at Waterloo the thing that's also going to be interesting is that because he has been so successful and basically almost conquered the entire continent of Europe the opponents have to get their act together and they work a little more on better coalition warfare and you're going to find that Waterloo when the Prussians get beaten before Waterloo instead of running away their king has told them stick with the British don't give up because divided we're going to fall united we're going to stand so this defeated Prussian army or at least part of it instead of pulling away to its logistic line the British to fight against a couple approaches that I'd recommend for Waterloo and I think there's a real accessible article in here the downfall of Napoleon at Waterloo by Brit a couple ways but one is what I like to do is have students give briefings take at the strategic the operational tactical level of war ask a pair of students one brief the Allied side one brief the French side strategic one brief the tactical another one way to approach this is to ask the question this is one that's kind of fun does Napoleon lose the battle or do the Allies win it kick that one around a little bit yet another way to approach Waterloo is say what were the key decisions made to determine the outcome of the battle of Waterloo and you can point to several one is the one you talked about the Prussian decision to fall back to link up with the British and they arrive in the nick of time to save the battle at Waterloo that's one key decision another key decision may be Napoleon's decision to send Grouchy after the Prussians he picks the wrong man Grouchy doesn't do a good job of chasing them and as a result the Prussians get away what are some other decisions you think are key the decision that he doesn't make no deception just a frontal attack if you look at Austerlitz there's some kind of things to force the enemy to come out well I think the enemies learn that he's not going to come out and play to his strength but he didn't try anything to deceive him didn't try a flank attack it was basically frontal assault and he left the wrong guy in charge for that I think that's a good point there too I think his personnel decisions are key in this too look at the cork commanders he has at Austerlitz none of these guys are going to play a key role at Waterloo the cork commanders who led the key attack at Austerlitz he's the chief of staff now he has to be well Napoleon decides he's going to give him that job because he's lost his old chief of staff D'Vu who held the key the right flank is the key player at Austerlitz is now back in Paris holding down the political situation as a minister of war because Napoleon is worried about his rear area so all of a sudden you've got a key cork commander now doing political military stuff Lon who is his key commander on the left flank at Austerlitz is dead he dies in one of these many campaigns that Napoleon is going to fight against Austria in this case in 1809 so the old cast of characters the old first string is gone here and he's playing with second stringers as cork commanders and division commanders and I think that's going to play a key role a smart coach in sports will retire after he's won a championship and not drag on and some people say that like 1808, 1809 right around that time already you start seeing really making mistakes with increasing number because it's only so long that you could be at the high level of insight energy, health he's a sick man he's a grumpy man he's a tired man he's a defeated man who's trying now to regain he has this urgency to win so he's pushed to maybe go to battle before he could play to his strengths now he has to play to the opponent's strength by getting a quick victory so he can take back home a victory and say, look it, I'm winning again because he knows the home front is questionable that's why he's left a good guy to watch it but he's also a guy he needed on the battlefield that is a great point and this is a point brought out in Peter Paray's article that you read early in this lesson is when Napoleon as emperor is also commander in chief you got some wonderful unity of command it allows you at the strategic level to move inside this decision cycling what does the downside of according to Paray and I think he makes a good argument Napoleon doesn't know any way to fight but total war he doesn't know how to fight a limited war the way that you might have in Frederick's time so he forces his enemy to fight a total war back against him there can be no negotiation with this guy you have to crush him because that's the only kind of war he fights because he unifies the political commander and the military commander together there's no checks and balances in the system either Napoleon the military commander wants to go out and fight the campaign the political leader said no no let's tailor this campaign to the political objectives we have together they only have one objective to go out and crush the enemy and in the end that's going to be a downfall there's no political restraint to this guy who's out there trying to win the decisive victory all the time one way that I like to look at Waterloo you might consider it is that you've got three different armies going to battle you've got the army of the 18th century in the British they're still very much a mercenary emphasizing marksmanship they haven't changed that much because their army just goes ashore stays close to the navy does its fight if it needs help it runs back to the fleet and sails away then you've got the army of the moment of 1850 in the French army a sick army, a tired army a badly bruised army that is not the army of 1804 1567 and then you've got force 19 you could bring that up appearing with depressions and that will help set the stage for the significant reforms that depressions are going to be making that will set a new standard in warfare replacing that of Napoleon but it will take until 1870 for everybody to really wake up and say this is now the first class army gone is the French army it's the Prussian army that's again setting the standard for warfare that's an excellent point it's a great drama watching Napoleon rise along with the French Revolution and then his ego start to undermine his genius and finally makes his downfall which is a dramatic case study it's hard to do better than the Waterloo I like to and I'll ask you for some closing thoughts one of the closing things I like to ask students about is covered in your program for joint education objectives on the title page lesson one of the objectives of the Joint Chiefs of Staff is for you to comprehend the relationship between the concepts of the Revolution of Military Affairs and the Military Technological Revolution the question is what the heck are they talking about what are the Revolution of Military Affairs and what's the Military Technical Revolution Military Technological Revolution in short by some definitions major technological developments come along and change the face of warfare let's say the appearance of the atomic bomb or the development of the Panzer Division Revolution of Military Affairs though is something of a great order of magnitude where the whole nature of war is changed now our modern definitions tend to focus on technology what drives that change again the atomic bomb if you want the appearance of the tank or the aircraft on the battlefield something like that major shift in the way war is fought we say we argue in the lead article the Revolution of War argues that a revolution occurred and Napoleon was the driver of it Napoleon was the key figure this begs the question what's the technology that drives that revolution the answer is it's not technology that drives that revolution at least that's my opinion I think there's general consensus that we could say that if you want he comes on the heels of a social and political revolution and uses those along with his genes Revolution of Military Affairs a whole new way of fighting warfare a way of war that involves the great masses of people, involves decisive bloody battles involves a level of violence again if you had a scale of violence for warfare where this was thermonuclear exchange and this was perhaps peacekeeping Frederick fought down here Napoleon took us up here now we're going to keep moving up the ladder as we go on and talk about how warfare was the 19th and 20th century Napoleon has taken as a quantum leap in the level of violence of warfare he's taken us a lot closer to the kind of total war we'll fight today it's a revolution of military affairs driven by technology, no basically he fought essentially the same weapons as his opponents absolutely, good points to sum up, one point I think I'd like to make that you suggested ways that you could approach Waterloo and get the students involved with little briefings of involvement by assigning one student to say okay what most significant change do you see taking place in the French army before Napoleon comes on the scene and argue why, and that takes them into the Napoleon period and how Napoleon uses that innovation or change, then what is the most important change that comes about a French revolution have another student do that and then what is the key to Napoleon's greatness as a person, what's the key to a system that he creates and assign each one, have them do a three minute summation of their ideas and that can generate some discussion, final thing is to kind of to sum up this thing about the revolution of military affairs is excellent if I may then just what are some of the significant changes if you could look at them, one is governments have changed, they're more democratic in spirit and means that they rely more on people power and you're going to have to make some political reforms to integrate the people in a different kind of way into government that leads also to changes in the nature and composition of armies, they're going to be more and more national armies with conscription coming from within society and not mercenary armies and they're going to be large in scope, you're going to have 200,000 man armies appearing on battles instead of the 13 20,000 that you saw in Frederick's time, they're going to be larger armies and what happens in strategy rather than sometimes avoiding battles battle becomes the centerpiece because this large army, you can't run away from it, it's not designed to go out there and demonstrate, a lot of money went into getting it into the field to fight so it's a battle strategy and the hope will be to get a decisive battle where you obliterate the army and get a political result, if you can't you do successive battles, keep going after the person until you defeat him so it has raised the scale of war, changed the strategy it's a battle oriented strategy with the goal to bring political victory through decisive victory on the battle field and it has brought in a new integration of combined arms onto the battlefield with the main killer now being artillery or before it was the rifle, you measured combat power by the number of rifles you had now it's a different kind of technique now and you're starting to have combined arms coming together, so war first change and Napoleon becomes the model of the great general who can fight in this era of increasingly larger armies


And here now there's different rules for generally accepted accounting principles on how you account for an investment. If it's going to be short term or mid or long term, I'm not going to get into details with that. I'm just going to give kind of the over the overview concepts here. So that's when we would have our short term investment on the books because now I'm still going to use that cash in the future. Fairly soon, you would think and then buy property, plant and equipment with it. Now if this was your personal books or you had a business that if obviously if you had a business that was in business for financial investments, investments in stocks and bonds, then you would be tracking investments in stocks and bonds in order to generate revenue from the stocks and bonds. But our business is not that it's a guitar shop like most businesses aren't investment businesses, right? Now on your personal side of things, you can also use QuickBooks for your personal investments. And in that case, you would be tracking of course your stocks and your bond accounts in QuickBooks and it works quite well to do that. The goal is just different for your personal than from the business. On the business, you're trying to use your assets to generate revenue in the future. That's you're trying to get a return on your assets from the business. On the personal side of things, your overarching goal is to live well or something like that, right? So because you because you were going to have expenses for vacations and stuff that have nothing to do with generating revenue. So it's a dip. But the double entry accounting system works in a similar fashion. So so that means then the question is, well, if I have the short term investments on the books, how am I going to break them out? Let's first just do a quick look if it was a personal thing. If it was on the personal side of things, you might want to group your short term investment accounts into say your financial institutions like Vanguard versus an e-trade versus your bank or whatever. If you have multiple places versus your 401k in your business or that you're employed at or something like that. And then you can adjust these periodically when you get the statements, which might be at the end of the month or year. Instead of trying to put every individual stock that you hold or even every individual mutual fund in here because you're going to get too much detail in QuickBooks. Remember that QuickBooks is not here to track the day to day transactions of your investments. There's other software that can do that. Oftentimes the websites themselves are the place to kind of track that day to day activity. If you're trying to make quick changes in terms of your stock positions on a day to day trading basis, but you want to be careful with that too as well. And you have other software like a personal capital, I think is one and quick end, I think is another that can give you the ending balances, your balance sheets information from the financial institutions as the market changes. But QuickBooks is not designed to do that. You could link to the banks and link to financial institutions, but QuickBooks doesn't want to give us the ending balance on the balance sheet. They want to give us the activity so that we can create not only the balance sheet, but also the income statement. So the point is it's kind of cheating to just take the ending balance and it's not just cheating. It's not going to help you out from a bookkeeping standpoint, although it'll give you a nice snapshot of where you stand. So you might use these different softwares in alignment. You might have a QuickBooks, you might have a personal capital that can help you to see where you stand in a snapshot kind of format. And also then you might want to break out your investments that are not under the umbrella of an IRA or a retirement account in short term. And then you might put long term assets as those that are under the umbrella of a 401K or an IRA or a 403B, those that you can't get into very easily so that you can see your liquid cash versus the cash that you can't really reach. So those are some concepts on the personal side. On the business side, we have similar kind of things. If I put the short term investments on the books, again, I'm probably going to try to put it in here at one lump sum investment in whatever the financial institution I have it in might have actually multiple investments. Meaning I might be in a mutual fund with multiple investments or I might have multiple mutual funds so I could break it out stocks versus bonds or just have one lump sum here that I changed periodically. Then I'm going to get interest or dividends on the investments. If that interest and dividends goes directly into my checking account, then I'm going to see them coming through the bank feeds and I might record them just simply with the bank feeds as they hit the checking account to income. So that means I would have income and I could go over here and I could then just have them increase income. Now normally if I was to increase income from an investments, then I would put it down at the bottom, not an income up top because it's not part of my normal operations. This business isn't there to generate interest and dividend income, but if we get dividend interest income, that would be great. I'm going to reflect it at the bottom just to show my net income before this other investment income would be the general idea. Now it's also possible for you to roll over your dividend income and invest it back in, which you would like to then make a journal entry once in a while to put it back into your investment in that case. So you have to record the dividend and interest and then increase your investment account. Meaning it would be a journal entry in essence increasing the investment, the amount that you're investing, the other side going to dividend and interest income. Now the other issue we have is that the investment account could go up or down in value due to the market, not due to dividends and interest. So it's just going to go up and down in value, the stock price for example. Now if we took the cost, usually when we think about a building down here for fixed assets, when a building goes up and down, generally accepted accounting principles in the United States usually says, well we're not going to go with the fluctuations of the increases and decreases and the market value of the building in part because we don't know what the market value is. There's other arguments than that, but a building is unique until you sell it, you don't really know what the cost is and people can manipulate the value of their books by having higher or lower appraisals on a building. But if you're talking about stocks and bonds that are publicly traded on a public exchange, then you can still have an argument that the fluctuations are not valid or whatever because you haven't realized them. However, you pretty well know what the value of your stock is because other stocks are trading for that same amount. Therefore there's a good argument to record your stocks and bonds if they're trading on a market at market value. But then the question is logistically how do I do that? If I get a report saying that my stocks went up, let's say by $500 or something like that, then how am I going to record that? I can increase this account by $500 or possibly I make another sub-account which shows the unrealized gain of $500 so I try to track it separately under the parent sub-account, another method you can use. And the other side, the two methods you could use is you could put the other side to equity, the argument being that you haven't yet realized it. So I'm just going to have the other side go to equity and not hit the income statement, but that's confusing to most people. So most people I would suggest then are going to report it on the income statement, not as income again, but down here as other income down at the bottom and just call it unrealized gains or losses. So those are the issues, just want to touch on them.


So, in the previous lecture we have started looking at the algebraic and the geometric multiplicity of an operator, right and we saw that there are some interesting features such as if you were to transform this operator or look at this operator subject to a new basis at least the geometric and the algebraic multiplicities do not change. In fact, we have also seen that the characteristic polynomial also remains invariant under a change of basis, right. So, today we are going to probe a little deeper into this idea of geometric multiplicity and subsequently we are going to come up with a condition for diagonalizability of a linear operator, right. So, we have seen that of pivotal importance in our studies has been this particular subspace, right where a is of course the operator or you can treat it as a matrix as I said we will not distinguish them too often here, right. So, this is the subspace that we have actually given a name we have called it W i and it is precisely the dimension of W i perhaps I denoted it by the variable l i if not mistaken is that what it was, right. So, this is what we call the geometric multiplicity of lambda i which is an eigenvalue of a, right. So, today we are going to see an interesting property of this W i, right. So, consider lambda 1, lambda 2 till lambda k to be distinct eigenvalues of a, all right. Then look at W which we now define as W 1 plus W 2 plus so on till W k we understand what this means it is a sum of subspaces we are familiar with this object, right. Of course, our V is finite dimensional again whenever we studying eigenvalues and eigenvectors V is finite dimensional. We have said already that if you have infinite dimensional vector spaces the existence of eigenvalues cannot be guaranteed. So, we will do well to keep that in mind. The point that is going to be made here is very interesting which is suppose B i is a basis for W i, right. So, W i's are of course these objects, right. Then the union of these basis sets is a basis for W, right. So, that is our claim that if you look at these subspaces so defined, yeah corresponding to distinct eigenvalues lambda 1 through lambda k no two of which are equal, right. And you look at those subspaces and because these are finite dimensional you will always have basis for these subspaces. So, you take the basis of W 1, take the basis vectors of W 2 and take the union of this set of basis vectors for each of those subspaces that will give you a basis for W, right. Generally the above is a direct sum. So, at this point I have not claimed that this is a direct sum it is just a sum of subspaces. But I am now claiming that this is also going to be a direct sum, alright. What is it that you immediately understand when we say something is a direct sum? The intersections of any two of those, yeah, is going to be trivial except from for the 0 vector nothing else can be in common between these objects. So, that is the whole claim, okay. So, if you want to prove this we will of course constitute in showing two things that is the first part the part prior to this direct sum will involve showing two things. What are those two things? The fact that this set this is a set, right. This is a set it is a union of multiple sets. So, it is a set the fact that this set that I box now is a linearly independent set and that this is a generating set for W. If I show that then I will be done with the first part, right. So, first I will show that this is indeed a generating set for any vector in W. That should be pretty obvious actually the part that about this being a generating set is not very hard to see. So, consider W belonging to W, alright. So, if W belongs to this subspace W which is nothing but the sum of these subspaces I can write little W as W1 plus W2 plus till Wk where Wi belongs to the subspace Wi I going from 1 through k. After all that is what it means for something to belong to the sum of subspaces. Now, where is this object coming from then W1 this object from W2. What do we know to be a basis for W1? The subspace W1 I mean its objects in B1, right. So, let elements in Bi be given by. So, remember again because it is finite dimensional it will have a termination point you cannot have infinite number of objects in this basis sets, right. So, let elements in Bi be given by Bi is equal to Wi1, Wi2 until Wi let us say what should I what variable should I use Mi which means that the dimension of Wi is little Mi, yeah, right. No questions about this alright which means that W can be written as summation alpha ij Wij of course there will be a double summation or maybe I will just open up the bracket and help you see this easily, okay. So, let us just say summation W1j alpha 1j going from 1 through M1 plus summation alpha 2j W2jj going from 1 through M2 until alpha what is the number kj Wkjj going from 1 through Mk, clear. These are objects in the basis B1 these are objects linear combination of objects in the basis B2 likewise these are linear combination of objects in the basis Bk. So, this then obviously belongs to the span of W11, W12 until W1M1, W21, W22 until W2M2 likewise until Wk1, Wk2 until WkMk, right. I could have used a double summation I just felt that this helps in better visualization which is nothing but in the span of what the union of Bi's isn't it, yeah. This is the union of Bi's I have just stitched together all the elements in individual basis sets and I have gotten this huge yeah Mk times k this large set right. So, this definitely is a generating set implies union Bk sorry union Bi I should write is a spanning set for W. So, the spanning part of this candidate has been verified now we need to ensure that this set this large set that we have here is also linearly independent right. So, I am going to erase this part. So, consider I am going to just use the same notation alright is going to be very handy and you will see in a moment why summation yes no we need to show that this set of vectors any vector that you have from a subspace belongs to the span yeah. So, that means this set actually does not contain anything more than W no, no, no. So, you might have some redundancies. So, for example, a basis for a original vector space is also a generating set for a subspace or proper subspace of the vector space you might just assign a weight 0 to those vectors in the basis which do not belong to that subspace. So, you might have redundancies, but of course, in that case it does not make sense because the objects themselves must also come from that subspace. So, in general of course, we do not encounter such situations. So, that is only trivially obvious the fact is that this is what we were required to show that this is generating set and we have done that. So, next part is to show that it is also linearly independent if it is linearly independent I put forth to you that this sum that I have written here yeah. So, that is why I did not I actually thought better of this than to use the double summation here because I am going to use this. So, if I now posit that this has to be linearly independent I have to first check that when this sum becomes equal to 0 it is only possible if each of these alpha 1 j alpha 1 1 alpha 1 2 till alpha 1 m 1 alpha 2 1 alpha 2 2 till alpha 2 m 2 and so on till alpha k 1 alpha k 2 so on till alpha k m k all of them have to vanish. If that is so only then I will be able to conclude that it is linearly independent. So, consider the same sum here except that the posing of this is a little different now. So, I am going to still use the let me use betas. So, I want to confuse you. So, let us say this is beta 1 j w 1 j plus summation j running from 1 through m 2 beta 2 j w 2 j plus so on till summation j running from 1 through m k beta k j w k j is equal to 0. Now, there is something very interesting that will happen forget about this part just focus on this we are claiming that this sum is 0 when is it possible? Now, if I just focus on this object where does this come from w 1 if I focus on this object this comes from w 2 if I focus on this object this comes from w k similarly. So, this means that no matter what this combination is after all at the end of the day this is just sum w 1 plus w 2 till w k that is vanishing right. So, this is my w 1 this is my w 2 and this is my w k. So, if this has to vanish then I must have a bunch of vectors each coming from those different kernels or different w i's whose sum must vanish, but what are these w i's after all? They are in the kernel of a minus lambda i i what sort of vectors live inside the kernel of a minus lambda i i the eigenvectors is not it. So, w 1 is an eigenvector corresponding to lambda 1 w 2 is an eigenvector corresponding to lambda 2 dot until w k is an eigenvector corresponding to lambda k and these are distinct eigenvalues. What do we know about eigenvectors corresponding to distinct eigenvalues? Lineal independent. Lineal independent and this is definitely a legit linear combination of linearly independent fellows. So, they cannot be 0 do you see they cannot be 0 unless all are zeros themselves. So, they are not even eigenvectors if they are eigenvectors then this is not possible. So, this is not possible unless w i is equal to 0 for i is equal to 1 to until k please ask if this is not clear this is the crux of the proof essentially please ask if this is not clear do not hesitate I will just repeat just in case you are missing the point here. The point is that this sum is just a basic test for linear independence right if it is linearly independent then each of these objects must vanish. So, I just put a linear combination of these fellows and equate it to 0 and check whether it is possible for any non-trivial linear combination straightforward very basics. I observe that this sum the first term of this comes from w 1. So, let us label it as w 1 the second term comes from the subspace w 2 let us call it little w 2. So, on the k th term comes from the subspace w k let us label it little w k and then I equate the sum to 0. The next observation is the crucial bit where I observe that after all objects inside the subspace w i are nothing, but eigenvectors corresponding to lambda i and the lambda i is by my very premise are all distinct and earlier much earlier in this course maybe some 4 5 lectures back we have proved that eigenvectors corresponding to distinct eigenvalues must be linearly independent and therefore, any linear combination you take of those eigenvectors that must be non-zero right any non-trivial linear combination. So, this is one such non-trivial linear combination because each of the coefficients is 1. So, what are we possibly missing the only possibility is that these fellows cannot be legitimate eigenvectors. How can we ensure that they are not legitimate eigenvectors? Well there is this one deal which says that the eigenvectors must be non-zero anything is inside any 0 vector is inside the kernel of any operator. So, only when the vector is non-zero non-trivial then it makes sense to define it as an eigenvector otherwise 0 will be the eigenvector for everything. No sense right. So, therefore, the only way that these fellows fail to be eigenvectors are if each of them individually is 0. If any one of them fails to be 0 then we cannot have this situation because then it is an eigenvector and linear combination of eigenvectors corresponding to distinct eigenvalues can never vanish. Unless the linear combination is trivial which it clearly is not is that clear. Now, if each individual w i is 0 that means each of these individual sums becomes 0 which means that summation beta i j j going from 1 through m i w i j is equal to 0. But these are after all what these are elements in the basis b i and a basis by definition is linearly independent. So, therefore, I arrive at the conclusion that. So, this is true for all i yeah. So, beta i j is equal to 0 for all i belonging to the set 1 2 until k and for all j belonging to the set 1 2 3 until m i. But that is essentially the definition of linear independence of these fellows which means that the union of the basis sets is also a linearly independent set right. So, let me erase this part now that means it is a linearly independent set it is a generating set. Of course, therefore, by putting together those 2 things now for the final bit which is when we have to show that it is a direct sum right. So, consider w belonging to intersection of w i and w j for i not equal to j which means that a minus lambda i i acting on w is equal to 0 a minus lambda j i acting on w is equal to 0. You take the difference and you are led to concluding that lambda j minus lambda i times w is equal to 0. But lambda j minus lambda i is clearly not equal to 0 because they are distinct eigenvalues yeah. So, therefore, w is equal to 0. So, anytime you have something in the intersection of any 2 of those subspaces it must be only the 0 vector and nothing more than that and therefore, indeed that was a direct sum right and you can also check by the dimensions now. Individual dimensions if they are m i then the overall dimension is summation m i that also tallies with our understanding of dimensions and how they work out over direct sums right. So, this is an important result which is going to pave the way for now our big result which is going to talk about diagonalizability of an operator. So, this part is clear if it is then I will probably it is that part and write our next result. So, for a which is an operator from v to itself of course, finite dimensional vector space let us just put it right there even if it is obvious ok. The following are equivalent 1 a is diagonalizable to algebraic multiplicity of lambda i is equal to dimension w i for all i ok lambda i obviously denotes the i th eigen value of a. So, of course, all that is implied here alright and third a has n. So, of course, finite dimensional let us just say the dimension is n that will make life easier a has n linearly independent eigen vectors. So, we are not talking about distinct eigen values or anything as it turns out we do not need distinct eigen values one clear cut example as I have mentioned earlier is the identity matrix all its eigen values are one they are all repeated the extreme case and yet it is a diagonal matrix. So, obviously, you do not require all n eigen values to be distinct it is a sufficient condition for diagonalizability as we have seen. Now, this is exactly or precisely the condition ok if you want diagonalizability then this is by the way what the geometric multiplicity of lambda i you may also write it as such right. So, I might have of course, defined the characteristic polynomial and shown you all that product form and all, but instead because I have already taken the trouble of defining the algebraic multiplicity I am just going to use that notation here yeah. This of course, is the definition of the geometric multiplicity this part right ok. What does it mean when we say that an operator is diagonalizable see the main crux of this will again be down to understanding what these statements actually mean once you see what they mean things will become very obvious. So, let us first try and understand or parse these statements. When we say that a is diagonalizable what we are essentially saying is that there is a basis in v corresponding to which the representation of the operator a turns out to be a diagonal matrix that is the meaning of diagonalizable. Remember right at the beginning of this course when we were talking about the preliminary goals of this course one was to solve a x is equal to b and the second was this p inverse a p getting it down to a diagonal form what is the p inverse a p now having done so many so much of this course having covered so much of this we have a better understanding that p inverse a p is just the representation of a subject to some change of basis some basis. So, if we can say that p inverse a p is equal to some diagonal matrix it means a is diagonalizable or in the language of basis we understand that there exists some basis subject to the choice of which the operator looks completely like a decoupled and we have seen the benefits of this in particularly in solving differential equations right you have nth order or nth degree differential equation it reduces to just solving n first order differential equations that is very simple is it not right. So, let us try and prove this what do you mean by unique basis I mean you take any vector you scale it up and down it is still an eigenvector you take any vector in the span of that particular vector right that is also an eigenvector is it not if 1 0 is an eigenvector of a particular 2 by 2 matrix then so is 1000 0 1 0 1000 0 1 billion 0 or 10 to the power minus 30 0 all of them are eigenvectors. So, I might choose a basis you might choose another basis you might scale up those matrices in different order in different manner there is no claim for uniqueness of eigenvectors eigenvalues sure unless you put more restriction that the eigenvector has to be of unit norm or something of such sort you will never be able to get it down to some unique object even so you might have multiple eigenvectors for the same eigenvalue for example, for the identity yeah you take any of the standard basis elements they are its eigenvectors but the funny thing is you take any vector in the Euclidean space it is an eigenvector so if you are already starting with the identity matrix you take any basis so such a basis is obviously not going to be unique right. So, there is no claim for uniqueness we will get into this question maybe after covering quite a bit we will get into this when something is diagonalizable and when things are not diagonalizable and so on and so forth and when they are not diagonalizable we will see the best we can do is Jordan form and then we will see why it makes sense to study the Jordan form because you might think it is a corner case which it is because unless you have repeated eigenvalues you never run into trouble see if you have distinct eigenvalues it is always diagonalizable you might think if you have repeated eigenvalues only then this seems to because and this also suggests as much so in general if you have distinct eigenvalues always diagonalizable so why shouldn't we stop our studies there and go further but sometimes the eigenvalues are not just about your numerical computation you see sometimes it is the nature of a physical system that dictates that your eigenvalues will have to be repeated suppose the physical model of a system necessitates that the system will have repeated eigenvalues it is not about your calculation or computation where you can just say oh just give it a little bit of tweaking and perturbation here and there and you immediately have distinct eigenvalues right so it still makes sense so we will now try and establish this any other questions on this okay maybe we will post on this proof to the next module