Material characterization course in the last two classes we reviewed about the electromagnetic lenses and its function, fabrication and some of the parameters which controls the electromagnetic lenses how it is being used in the electron microscope. A kind of introduction with little more details we have gone through. Now from this class onwards we will just start the scanning electron microscopy where all this electromagnetic lenses we have seen will be used. So before I just start this lectures on fundamentals of scanning electron microscopy I would like you to carefully go through what is shown on this slide. So before we get into any of this electron optics or electron optics based instrumentation which is used for imaging of materials to reveal the microstructure details first we should know about the interaction of an electron beam with a solid. It is a very gentle information which one should remember I will tell you the importance of this the moment I finish this discussion here. So look at this schematic what is shown in this slide is you have a specimen and then this is an incident high energy electron beam which is falling on the sample and then you get to see quite a bit of signals which is coming out of this sample in all the directions. So I would like you to carefully look at each one of them. So what we are seeing is within this volume of the sample what we have written is an absorbed electrons some that means some electrons are being absorbed by the specimen and some of them actually you get electron hole pairs generation and then you see a secondary electrons elastic x-rays visible light and then you have back scattered electrons and then you have elastically scattered electrons and then you have a direct beam and you have inelastically scattered electron and then you have Braemstrahlen x-rays. So by looking at this you just see that when a high energy electron beam interacts with the specimen it is always true that all these signals are generated. It is this the detecting system which you employ to collect them and use them for imaging or analysis that characterizes the particular characterization equipment. For example you just see that the visible light we used so for an optical microscope a characteristic x-rays can be used for n number of spectroscopic techniques to analyze the chemical details or chemistry of the specimen in a very very high resolution of in the materials which are I mean which may contain very minute or trace elements. So in order to characterize them we may use this characteristic x-rays. We will look at that all the spectroscopic technique in a different lecture series but you just see here this is also one of the important signals which you get out of the electron beam specimen interaction and then this back scattered electrons and secondary electrons are being used in SEM and then you see that OJ electrons are used in OJ electron spectroscopy and then a direct beam which comes from the specimen is used in transmission electron microscopy and you have all this other signal also being used in the transmitted electron microscopy for different applications we will look at that in an appropriate time. So I just want you to look at all this signals which is coming out of the specimen these can be broadly categorized into two segments. One is a forward scattering signals all these are just direct beam inelastically scattered electron the elastically scattered electrons these are all forward scattering signals and then you have a backward scattering signals. So out of these two categories the scanning electron microscopy uses only the backward scattering signals. So this is primary important information one should have before we get into the details. So all the other signals are not used in the scanning electron microscopy we will see the details one by one but as an introduction you should know in general when an electro I mean high energy electron beam interacts with the specimen all these signals are coming out and then the kind of detecting system which we use actually defines the characterization tool whether it is a scanning electron microscopy or a transmission electron microscopy or any spectroscopy specific spectroscopy which where we look at the chemistry of the specimen. So this is primary important concept you have to understand before we get into the specific characterization tool. So with this introduction I would like to start the scanning electron microscopy and let me just go to the blackboard and then write few things and an introductory remark. So in an introduction to the scanning electron microscopy we should know what are its unique capabilities why do we opt for in a scanning electron microscopy investigation in comparison to a light optical microscopy. The primary objective is to obtain the magnification with a high resolution. So in a very simple terms you can obtain microscopic details 3D like images you will see how this effect comes and what are the parameters which contribute to this phenomenon or an effect I would say. Magnification in the range of 10x to 10000x and more in fact it could be more also. Typically the signals are obtained from the specific emission volumes within the samples and can be used to examine the sample in terms of surface topography, crystallography and composition etc. So these unique characteristics we could not do with a light optical microscope and what is surface topography the surface unevenness. So you can just imagine what we have seen in a light optical system if you recall we just polished the metallic specimen with the different kinds of emery sheets right. So the final emery sheets which had very fine ceramic particles embedded in that sheet and then we just rubbed the sample against them and then that sample appeared almost like shiny and so on with our naked eye the sample looked very polished and so on then we what we did it we also put that sample under the optical microscope then we could observe the very very closely spaced scratches. I would say this closely spaced and impression which was observed like if you put the same sample under the SEM you will see that there are hills and valleys because we are looking at the very high magnification rather a high resolution we are looking at it we are able to observe the small hills and valleys that is surface topology. So this is one classical example you can just go and look back this can be so any surface unevenness to the very micron to nanometer scale can be analyzed and also the crystallography of the specimen and its chemistry can be analyzed with the scanning electron microscope primarily. So what are the parameters which enables this microscope to do that you will see that so what are the typical signals we are going to get from the SEM secondary electrons backscattered electrons x-rays and other photons of varying energies just you look at that slide again what I have just shown here this is secondary electrons backscattered electrons characteristic x-rays and other photons of varying energies see each radiation will have very specific energies which we will talk about. So the primary signals which is coming out of this SEM as I said it is a backscattered signal or backscattering signal sorry I would say backwards scattering signals it will be very clear because there is another particular signal is named as backscattered electrons so you should not confuse with this because this is only coming the backward you know scattering this is what I meant all these signals are backwards scattering signals which are being primarily used in SEM. So only these three signals are primarily used in an SEM of course they are characterized based on their energy that we will see in an appropriate time and out of all this backwards scattering signals only we talk about is secondary electrons and backscattered electrons why why are we talking only about this because they vary primarily as a result of difference in the surface topography. The amount of secondary and backscattered electrons which is coming out of the specimen surface is primarily depending on the surface topology. This is the core idea behind using this too and what are the other important things so the other important information you get from the SEM is the x-rays that is characteristic x-rays come from a sample can yield both qualitative and quantitative information from the region of 1 micrometer diameter and 1 micrometer depth. This is a rough indication you get what is the region size from which you get this information which is of in the order of 1 micrometer diameter and 1 micrometer depth from the surface. So these are the information you get from this in general from SEM and we look at the what are the imaging capabilities. So if you look at the imaging capabilities of this microscopy the major reason for the SEM's usefulness is the high resolution in the order of 1, 2, 5 nanometers and another important feature of SEM is the large depth of field. We have already discussed in the fundamentals of the optics we have seen what is depth of field how it is being exploited in electron microscope. In fact the what we have just stated in the beginning 3D live images it is partly because of this effect you have high or very large depth of field in an SEM. We will also see it using a ray diagram how it enables this effect when we discuss the other functions of SEM's in the coming classes and the SEM's are becoming very popular because of the advances in the signal processing and amplification like the kind of signals you receive second electrons, back-to-skirt electrons or X-rays and then you have advanced processing signal processing and amplification detectors and then gun design etc. So with that all this advances this SEM's can also do imaging something like using electron channeling contrast by varying the crystal orientation and also magnetic contrast from the magnetic domains in the uniaxial and the cubic materials. So these are all some of the highlights of the imaging capabilities of the SEM. You will see that next is structural analysis. If you look at the what are the structural analysis one can do with this SEM it has got a capability to determine the crystal structure and grain orientation of the crystals on the surface. Please understand you have to remember that it is all what are the information you obtain is only from the surface with very limited volume. You will just understand that much more detail as we go into this lectures and then the diffraction of the back-skirted electrons emerging from the sample surface electron back-skirted diffraction EBSD with a low intensity also enables this capability and since it is a low intensity we have very high sensitive CCD camera recording. This is a charge coupled device camera records the back-skirted a so-called Kikuchi pattern which is nothing but this signal and it is analyzed with the computer based indexing method and then you have today SEMs with advanced indexing and computer assisted crystal lattice orientation mapping that is called EBSD maps which allow this technique to identify the phase and the misorientation across the boundaries. So this is also very powerful technique today and it has been applied everywhere in this itself separate research domain people can extensively use this and very powerful technique as far as SEM structural analysis concerned. And what else we can do with the SEM? So far we have seen imaging capabilities structural analysis and finally elemental analysis. So if you look at the elemental analysis capability of an SEM you can get the complete compositional information using characteristic X-rays. The tool generally referred as electron probe micro analysis EPMA which can get the chemical composition from the very localized region and then provide complete chemical analysis and then you have this EPMA especially outfitted SEM with light optics and one or more WDS units wavelength dispersive spectrometer. We will see all this variants of the spectrometers as I mentioned which uses the characteristic X-rays which comes out of the sample and then do the chemical analysis. We will look at them in a separate lecture series but then these are all the attachment one of the primary attachment to the SEMs. This one the another one is energy dispersive spectrometer which can detect the elements greater than 4 atomic number collect the characteristic X-rays from the major elements approximately should have about 10 weight percent whereas the WDS measures X-rays from the minor or even trace elements of 0.18 percent. So WDS is much more powerful compared to EDS. We will see why and so all these details later but these are all the basic details one should have about the when you look at the capabilities of an scanning electron microscope. I have few more points to add in this segment. So the final point to the elemental analysis with a modern EPMA you can get a quantitative information from your specimen within a spatial resolution of the order of 1 micrometer with the accuracy of the order of 1 to 2 percent the amount present and also this EPMA has a capability of analyzing the very low atomic number elements like boron, carbon, oxygen because the WDS spectrometer uses a large interplanar spacing diffractors typically organic crystals which has got large interplanar spacing which enables long wavelength X-rays from the low atomic number elements. Since this low atomic number elements has the characteristic X-rays of large wavelength or long wavelength. So these crystals enables the diffraction possible and then and they can be measured with the WDS. So these are all the the basic capabilities of a scanning electron microscope. So I have just put them into three categories one is imaging capabilities and structural analysis and then composition analysis or elemental analysis and ACMs are primarily used for only this purpose. Now we will look at the some of the other introductory remarks as the sophistication of the investigations increased the optical microscope often has been replaced by instrumentation having superior spatial resolution or depth of focus. So the resolution of ACM can approach a few nanometer as I mentioned and it can operate at magnification that are easily adjust from about 10 X to 300,000 X. Of course this can be a subject of instrumentation we will talk about it in an appropriate time. The depth from which all this information comes varies from nanometers to micrometers. This also a subject to a specific instrumentation details we will look at them in appropriate time. Likewise the lateral resolution in this analytical modes also varies and is always poorer than the topological contrast mode. So the principal images produced in an SEM are of three types namely secondary electron images, backscattered electron images and then you also have elemental X-ray maps and these are the three primary images one can obtain in a normal ACM. Secondary and backscattered electrons are conventionally separated according to their energies. So we will now see how this will have some schematics to show how this ACM works. So you have the two separate entities one is a microscope column from the top to bottom the other is a control console. So you have the electron gun which comes from the top of the equipment column microscopic column and then you have further all the electron lenses and a scan coil and the beam reaches all the way up to this chamber specimen chamber which is maintained at the with the vacuum of 10 to power minus 4 Pascal which is in the order of 1 billionth of the atmospheric pressure for the for your reference. And you have on the right hand side you have the CRT screens viewing screens and then camera where all this scanned images are being used. So this is a primary classification of this equipment microscopic column and a control console and then you look at this next schematic and you have this a complete schematic of the cross-section of the scanning electron microscope and what you see is an electron gun which generates the electrons and accelerates to typically from 0.1 to 30 kilo electron volts and then this being passed through electron lenses also and also a scan coils. So these electron lenses what they do is the probe diameter which is being produced by this the junction harping typically it is not sharp enough to resolve the structures and these electron lenses de magnified to very sharp spots or a sharp probe and then they are being rastered on to this coil to the this specimen and then you have this detector the signals which comes out of this specimen which is kept under the vacuum and then it amplifies and it goes to display. We will continue the discussion of this general function of this scanning electron microscopy in next class as well. Thank you.


All right, we're about to go rogue from our pattern here We can no longer fit everything you need to know on this One page we can fit our summary on here the how does this happen? We're gonna have to get a new page and we're gonna have to draw it out and talk it through but the summary We've already got the electron transport chain results in ATP Does anybody want to guess how many ATP molecules do you think we get out of the electron transport chain? And I'm going to say this probably as like a There's a range there's some variation in how much and so the number I'm gonna give you is like an ish It's about this much ATP not exactly maybe a little bit more maybe a little bit less And that's fine. I'm not gonna be concerned about exact numbers, but make sure that you know what your Instructor is wanting from you because Sometimes they give you exact numbers and then they they want those numbers. I Wish you were all here so I could hear what your guesses are like are they super high or are they super low? Do you think like five ATPs are coming out of this thing? 10 thousand Not a thousand 28 28 ATPs Almost three times more than we've gotten anywhere else in the process Two times more our total almost number of ATP that we have generated through the whole other process 28 ATP are gonna come just from the electron transport chain What are the other things that we've been keeping track of in the whole process we've been going where does this happen? We're gonna have to draw a picture in order to figure out where this whole thing is happening and then we've been keeping track of the number of enzymes involved and I'm gonna say that there's one enzyme involved and a pile of Transporters, but you are carriers and you tell me if you agree with that. I I'm like Trying to put it into a box, but you can tell me what you think of that Okay, so where where is this happening? Well, oh wait, let's fill this in we Produced no additional electron carriers, but like I said we got 28 ATP molecules No carbon dioxide, but go ahead take a wild guess. What do you think is gonna happen with water? We're gonna get six water molecules out of this thing and I'll tell you right now There's oxygen involved because we haven't seen where we used up the oxygen that we breathed in We haven't figured out like what happens with the oxygen oxygen is super important Here's where this happens I'm going to embed proteins in the inner mitochondrial membrane and This is the electron transport chain now I Can't see those and that's like not even close to being big enough for me So I'm gonna have to draw this out much bigger But that's our location This inter membrane space is an important part now, I'm going to take a little like I don't know a snapshot of this piece of Mitochondria and I'm gonna blow it up so that we end up with our outer Mitochondrial membrane are you cool with that? Which means out here is the cytoplasm of the cell and I'm gonna include the inner mitochondrial membrane which means this is the inter membrane space Which is a really important place In that inner mitochondrial membrane. This is where we had those proteins that are Embedded and I'm gonna draw them totally diagrammatic my friends This is not what they look like and in fact, I think I might have a Picture of what they do look like somewhere around here Nope, I'll have to go find it for you so I don't make you dizzy looking for it I'm also gonna go ahead and just move my label My inner mitochondrial membrane label are you cool if I just move it down there so I can draw more stuff And I'll Point to it like that Because there's one other thing I want to draw I Want to draw our single enzyme and I'm gonna draw our enzyme like this There's a reason I'm drawing it like this Okay, this enzyme. I'll tell you what its name is first. I was trying to figure out like how I'm gonna make this story so exciting it's ATP synthase ATP synthase is an enzyme embedded in the inter mitochondrial membrane and Maybe take a minute What do you think? ATP synthase is Going to do what is this enzyme gonna do? Well, I'll tell you it takes a DP plus P and It turns Them into what do you think? ATP this is how we get our 28 ATP molecules ATP synthase embedded in that inter mitochondrial membrane Makes it happen makes it rain Thank you very much ATP synthase From ADP plus P. Where did that come from? All the work that you're doing in your cells all the transporters that need ATP to make stuff happen They're all using up a molecule of ATP and what results is ADP and P and those ADP and P's Travel to the ATP synthase go through the process and become ATP once again. We're just recycling this stuff Because matter can't be destroyed. We're just gonna reuse it We're just gonna take the little Lego pieces and make something new because ATP is like gold So if we can make more of it good gracious, this is an excellent idea. Let's do that Okay, who are the players that you would go wait a minute how well like what how is this even possible? Who's providing energy For ATP synthase to do the work of building ATP because you know that's gonna require energy, right? Right, of course because you're gonna have to rip the P off of a water molecule and Then somehow stick it onto the ADP in order to get like kind of an unstable ATP molecule So where does the energy come from? Well, we just spent a whole lot of talking time looking at Transferring the energy from the glucose molecule into our electron cars all 12 of them are gonna come and What do they bring in? They're bringing high energy electrons to the electron transport chain It's about to get wild Because watch what's gonna happen The electron carrier sidles up to protein number one I don't like where that one is placed These are not their names. These are not what they look like and I'm simplifying just so we can grasp this weird entire process Along come the electron cars Bringing their electrons their high energy electrons and they transfer the high energy electrons To the proteins Now here's how I envision it. Here's how I imagine what's happening The electron carriers have high energy electrons. They pass the electron to the first Protein that first protein grabs those electrons and in that transition in that passing There was some energy released It's almost like those high energy I mean really I think of it this way that we have these high energy electrons that are high like they have a lot of potential energy and then we pass them downhill and More energy is released They're not really being passed downhill, but this is just the way that my brain visualizes like how is this possible? It's all electron energy Awesome. I trust that But the visual of passing them downhill I can be like, oh, there's energy there. I see that now What are these proteins going to do with that energy? It has to do with ATP synthase and its function Might seem weird to start with They use the energy. Oh My god, this is so cool. They use the energy to pump Hydrogen ions or protons into the inter membrane space So a question where does the energy come from to pump? Hydrogens in hydrogen ions into the inter membrane space It came from the high from the high-energy electron carriers Where did the energy come from that was in the high-energy electron carriers? It came from the glucose molecule. Oh Where did the energy in the glucose molecule come from? It came from the sun That's the next story that we're gonna tell in the next lecture in the meantime all 12 electron carriers show up to this first proton protein and Say here's some energy Take these electrons pass them along and all they pass them along and they create a Hydrogen Ion gradient Now You will see this Also referred to as a proton gradient and if you think about it a hydrogen ion a positively charged hydrogen ion is the same as a single proton So either way you want to say it is fine. I don't know which way I say it more often I think having the little H there it I'm describing it as hydrogen And I think that for me makes it a little bit easier. So now I've created a High concentration of hydrogen ions inside a lower concentration of hydrogen ions outside Do you are you seeing any hints? Are you like whoa? Wait a minute. We're building into a story What it we've created a concentration gradient and If you were a hydrogen ion and you just got pumped into the inner membrane space with like jillions of other hydrogen ions all around you What are you gonna want to do? You're like dude get me out of here and in fact Guess what your only exit is is Every topic the best topic in the whole world like how amazing is this? The only way out for these hydrogen ions is through ATP synthase and when they go out ATP synthase I'm not exaggerating or joking. There's like a little wheel in here and They spin the wheel And they leave and The little wheel inside the ATP synthase Now that's where the energy comes from The ATP synthase takes the energy captures the energy in that little spinning wheel that came from those hydrogen ions Rollin through there because they want out and Uses that energy to just to make ATP 28 ATPs as a matter of fact Now has anybody anticipating we haven't gotten to an important part of the story Anybody anticipating a problem here? Well, here's the problem Those electron carriers can only hold two electrons I mean the chain the the electron the electron carriers can only hold two but the proteins in the electron Transport chain can only hold two electrons at a time so if we had all of these Electrons here all these proteins have electrons just like I visualized here our Process backs up actually do you agree with that because look our electron car shows up It's got it's like dude. I've got electrons for you and that first protein is like Somebody take these because I there's more down there. I'd like to get those but I don't have anybody I can pass them to Guess what oxygen does? Take a deep breath and thank your oxygen oxygen is the final electron acceptor and You the whole thing would back up all the way to glycolysis If you didn't have oxygen as the final electron acceptor and if the whole thing backs up You're not gonna get 28 ATP out of this process Oxygen comes in accepts those final electrons Guess what it turns into True story it turns into your water molecules The oxygen collects a couple of electrons collects a couple of hydrogen ions Puts it all together and voila now we have a water molecule If we go through the whole process we'll end up with six water molecules. Where is my? List oh look there I already put it in there that our six water molecules come out of the electron transport chain Do you feel like? Oh, that's perfectly clear. It makes complete sense, of course It's definitely a story and in this next Piece of this lecture. I'm gonna go through a little animation visual that I created That's review if you look at this whole thing in your life home kid I do not need a review right now. I can totally visualize what's going on. I've got the pieces down I've got the crazy cool diagrams down. I'm ready to go Skip the next section, which is the animation review But if you want a visual review of the whole thing from start to finish That's what we're gonna do next We're not done After that review, we're gonna look at Aerobic like we're gonna look at the consequences of not having oxygen and then we're gonna look at some other things that can feed into Cellular respiration not just carbohydrates not just glucose we can use proteins and fats as well


So, the next claim so B prime this is linearly independent once again when we map something to 0 it is very important that you understand what 0 are we talking about the 0 of which vector space this is the 0 of the vector space of linear functionals. In other words to show that summation alpha i f i is equal to 0 v yeah implies alpha i is equal to 0 for all i that is what the linear independence condition boils down to. Yeah of course every field must have the 1 and the 0 right yeah it is 1 and 0 of the field because it is getting mapped yeah if it is all 0s then it is the unimportant like it is the boring case when it is a 0 functional right ok. So, what are we saying here we are saying that ok yeah so this 0 of this v prime yeah so the 0 of the v prime means that every object this means that summation alpha i f i of course i going from 1 through n i going from 1 through n this v is equal to 0 where is the 0 of is the 0 of the field now because I have passed on the argument now. So, this is not in the sense of the functional anymore so this is the 0 of the field for every v belonging to vector space v and that is the most important observation that is it has to be true for every v if it is true for every v I can go ahead and choose some very special v's and conclude certain things about this yeah so choose v is equal to v 1 say for instance and what do I immediately have this means that alpha 1 f 1 v 1 is there anything else really to it because when you take f terms f 2 onwards f 2 acting on v 1 is what 0 so it contains no other term. So, the only thing is this, but what is f 1 v 1 that is 1 so this means that alpha 1 is equal to 0, but is really 1 any special or any more special than any of the other numbers from 1 through n. So, I can just you know choose this i maybe I will take the shortcut while I expect you to repeat this with other values, but I will just take this for any i yeah. So, any i you choose you are led to eventually concluding that alpha i must be 0. So, this means that alpha i is equal to 0 for all i which is exactly what I wanted to show. So, therefore, you have a linearly independent set you have a generating set therefore, by virtue of those 2 claims the big result is that is why I have not erased that part this b prime is a basis for v prime not just that you will see that we call it b prime is a dual basis to b. So, given any basis in the original vector space v I have a corresponding dual basis by virtue of this Kronecker delta thingy here right. I will just work out a very simple example in R 3. So, this is clear right this idea is clear the fact that it says generating set and a linearly independent set and therefore, a basis for the dual space what is it that immediately strikes out the dimensions of v and v prime must be equal therefore, they are isomorphic right, but maybe not in today's lecture the next lecture we shall see there is something very interesting in order to show that these two are isomorphic you had to actually construct basis. So, it is said that there are levels of proof that involve constructions of basis and then there is a higher level of proof which does not go through this basis construction. So, it is like you know it is a good time to read George Orwell by the way. So, there is this Orwellian line that all animals are equal, but some are more equal than others right. So, you have vector spaces that are isomorphic, but some vector spaces are more isomorphic than others and it turns out that there are objects such as double duals. So, you have the vector space v you have the vector space v prime which is a dual to v and then you have a vector space v double prime which is a dual to v prime and it turns out that v double prime is also isomorphic to both of these aforesaid spaces, but in order to show the isomorphism between v and v double prime that is a double dual you do not need to go through any construction of basis. They are what we call as natural isomorphisms. So, isomorphic vector spaces are of course equals in some sense, but in order to prove the isomorphism between v and v prime we had to go through this route of constructing an explicit basis for the second vector space in terms of the first there is this dual v prime as a dual to b when in the next lecture we return and show you that the double prime or the double dual is actually also isomorphic to v we will not require the choice of any basis. It will be a natural isomorphism in some way it might appear a little mystifying at this point, but we shall see that they are actually more equal than this first and not just that if you just expand your sort a little bit you will see that this double. So, the double prime then the fourth prime then the sixth prime and all these happen to be more naturally isomorphic and the first dual the third dual the fifth dual onwards they also happen to be naturally isomorphic I mean it is like some shadow spaces that are happening right their isomorphisms. So, all of them you keep taking duals they keep becoming isomorphic to one another, but that the one hop the skip if you like first to the third or the v to the double dual they happen to be more naturally isomorphic. We will see that proof in the next lecture, but today we shall continue with this and see something interesting as well. So, what is it that we have seen that there is a double there is a dual basis to every basis that we can cook up in the original vector space v. So, let us take the simplest example one can think of. So, let us suppose v is equal to R 3 and let us say b just illustrate the idea b is equal to say 1 0 0 minus 1 1 0 and 1 minus 1 minus 1 I leave it to you to check that this is a linearly independent set the cardinality is 3. So, it is indeed a basis right. So, check to see that b is a basis for R 3. So, by the way what do you think a dual to R 3 would look like? Any guesses what sort of objects will take will ingest objects like these and spit out a member in the field that is a real number. So, if you take this as column vector I put it to you that you take this as R 3 row vectors that is all that is going on all this while we have been talking about this dual, but if you view it in terms of coordinates it is really nothing, but the action of some row vector on these column vectors that will spit out of scalar in the field right. So, now just to explicitly. So, of course I am assuming that you will check and convince yourself that this is indeed legitimate basis for R 3 and once we have the fact that this is the dual space the row vectors is what we are looking for. Any questions here right here? So, now all that we need to construct is a basis for those row vectors those vector space of row vectors 3 tuples right. So, what do we need? We need firstly an object which when we take multiplication with this spits out of 1, but with any of these other fellows it must spit out 0 remember it is a chronicle delta right. So, what must it have over here? See first of all with this and this it must be 0. So, if I am looking for the first object here let us see b prime if I am looking at the first object here my first observation is that the first two entries must be equals should not otherwise this cannot vanish. See when this object this is f 1 acting on v 2 must be 0 right. So, therefore the first two objects must be identical right. What about this? The third object maybe I can just put the third object as 0 it matters not. The first two objects what should I put? Just 1 and 1 check this times this is just 1 this times this is 0 and this times this is also 0 right. What about the second object f 2? I need something that takes this and makes it 0 here, but I need to have 1 here. So, what should I do? And I need to have 0 here again. So, this object I need to make it 0 and these two objects are somehow also badgering me a bit. So, what should I choose? Sorry 0 1 minus 1 exactly right. So, you take for instance this 0 this picks out this and this filters out these two, but their difference is 0. What about the third? So, this is actually the dual basis that I am constructing here. Now you can think of arbitrary abstract vector spaces and you proceed in a similar manner and you should be able to construct duals. If there is a problem just go ahead and go to their coordinate representations that will be simple enough right. Because we have taught you that finite dimensional vector spaces take them to their basic coordinate representations and that is all that you need to understand them. So, what is this? Just to complete the story? 0 0 minus 1 yeah. So, this is indeed the dual basis to this yeah what we have just learned. Any questions on this so far? This is just the example right ok. Now let us look at the following situation v finite dimensional vector space over F b is equal to v 1, v 2, v n is a basis for v. Suppose w yeah I mean I could have started the other way round I could have started with a w taken a basis for w and expended it to a basis for v it does not matter you get the idea. Suppose w which is a subspace of v and subspace given by w it is span of say v 1, v 2 till v k right of course, k is less than or equal to n. So, w is of course, a subspace sitting inside v now we will define a new object w with superscript 0 yeah is defined in the following manner F belonging to v prime such that F of w is equal to 0 for all w belonging to w. So, this is called the annihilator of the subspace w ok why because every time you feed it some object from w yeah every time you feed it some object from w it just takes it to 0 nothing much to read into it except for the definition as of now. Now the first non-trivial claim that shall be made is that this object this annihilator is a vector space of course, over F. So, let me write that in brackets lest it create confusion ok how do we show it is a vector space well you take any two objects F 1 and F 2 I will just a very sketchy proof of this alpha F 1 plus F 2 right if this object where to act on some object picked from w then what should it be because of the linearity we can write this as alpha F 1 acting on that object plus F 2 acting on that object, but each of them individually is a 0. So, therefore, this is indeed the 0 of the field and we are done right. So, the annihilator is indeed a vector space, but the more interesting claim is. So, if your original vector space w which is a subspace of v is spanned by the first k fellows out of this n in the original basis then go ahead look at the dual basis all right. So, maybe I should say that v prime is equal to span of F 1 till F n where F i v j is equal to delta i j like before. So, now if the first k fellows out of this n are a spanning set or a generating set of course, they are linearly independent any subset of a linearly independent set is linearly independent we prove that. So, if this is a basis for w then the last n minus k fellows of this dual basis is a basis for the annihilator if this is true if this is true what can we say about the dimensions of w and it is annihilator yeah the dimension of w. So, if this is true the consequences the consequences that the dimension of w plus the dimension of it is annihilator is equal to the dimension of v and that is precisely I urge you to think about this on for a moment nothing but the rank nullity theorem just think of it in terms of matrices or other things and you will be able to precisely see what I am what I mean by that, but the point is that we have to show that this is a basis is linearly independent under question of course, not because this is a subset of a basis. So, it must be linearly independent the only thing that I need is to show first that this is contained inside span of f k plus 1 till f n and the fact that this span of f k sorry f k plus 1 till f n is also contained inside the annihilator that is the two things we shall prove in order to establish that there is a basis for w naught or the annihilator of w right. So, we will do a detailed write up of this, but I will just give you the idea of how and why this carries forward maybe that will help you grasp the proof better in the next lecture. So, what do we have to do in order to prove that something is spanned by this you take what is the defining property of objects inside this you take any typical fellow sitting inside the annihilator of w it means that that linear functional must pulverize every object inside w. So, when that linear functional acts on fellows which are linear combinations of the first k fellows here it must map it to 0, but look after all every object that is every object that is sitting inside this w naught is coming from where it is coming from v prime remember v prime just like w is coming from v w naught the annihilator is coming from v prime. So, in general an object inside v prime has some special properties and then it belongs to w annihilator. So, any object inside v prime is a linear combination of objects inside this set. Now, if you pass on the special fellows that are residing inside w then they must be pulverized right. So, if you pass on very special fellows like this v 1 v 2 v 3 what do you immediately see if you pass on v 1 to a linear combination of these fellows right or the other way round if you pass on v 1 to a linear combination of where is it the f's yeah these fellows what do you immediately conclude that the coefficient of f 1 must be 0 the others are anyway 0's, but alpha times f 1 v 1 must be 0 otherwise it cannot be an annihilator right. Similarly, for f 2 v 2 therefore, the coefficient of f 2 must be 0. So, when you are saying that any object in v prime for it to also be an object inside w 0 the first k coefficients of these f's must all vanish and therefore, the only non trivial coefficients that will be left with are those of the last n minus k. So, therefore, this is indeed something that is spanned by or generated by these fellows and now it is only trivial to show that if there is any object that is spanned by this and you feed any object from w to these fellows all of them identically vanish because it cannot contain components from v k plus 1 onwards. Any object that is residing inside w is a linear combination of objects from v 1 through v k. So, when any of those objects from v 1 through v k this is the indicator function as your friend rightly said. So, when this indicator tries to check for objects from v 1 through v k they do not light up they are all 0's anyway. So, this is also contained inside this we will write this in more formal terms do not worry about it, but the point is that we will show that this is contained inside this this is also contained inside this and therefore, they are nothing but equal. So, therefore, the dimension of the annihilator given that the dimension of the subspace is n minus k is going to be n minus k where n is the dimension of the vector space. Now, you think about which is the image and which is the kernel and so on and so forth, but I put it to you that this as a consequence of what we shall prove next day is another statement of the rank nullity theorem just put some thought behind this that is the important concept to take home from this. Thank you.


Clara did love Peter.  But she was undecided about whether she wanted to marry him. He had been out of school for four years and still hadn't found a job. She had tried to stay away from the subject, but he was proposing marriage. That had brought it forward in a way she felt compelled to answer. She had tried to set him down easily. I want a home and a family, but we can't make a life together on love alone. Economics is a part of life. The very purpose of family is merging into a unit, and that commonly comes with the human intent to bring about a next generation. People without an income have very restricted opportunities. They have less to offer as their part in the nation and less to offer in support of a family. In the United States, we have a sovereign people who own their nation, who own their government, and who own their economy. It is owners who set purpose to what they own. The purpose for the U.S. economy is the purpose of the people. It is the purpose that includes economic freedom and prosperity. And this perspective is essential. It is our key for effectively understanding our economy. People are the only party in interest. Nobody else matters. If the economy is not serving the people, it does not serve anybody. There are only two real performance metrics for our national economy. One is what goes to people as economic product. The other is what comes from people, what they must commit to get what the economy produces. There are only two real performance metrics for our national economy. One is what goes to people as economic product. The other is what people must commit to get what the economy produces. The human purpose for having an economy is converting the time and effort we choose to commit to productive pursuits into the goods and services that we both receive and value. This is also the citizen purpose. But we are the people who commit time and effort to productive activities. It is also the public purpose, for we are the public. There is no other public to receive what our economy produces. The question isn't how many people are unemployed. It is whether our economy has needful employment for people to earn good income from their efforts. Performance of the economy is in freedom to choose where to work and prosperity that comes with what people commit to economic activities. It is citizens being able to choose to spend on what their economy produces to meet their needs and wants. It is economic freedom. It is having sufficient resources to live well, to raise the next generation and to support senior family members in retirement. When someone else decides where you work, it is economic slavery. Having access to only the products that others would have you purchase is economic slavery. Having someone else spend your money on what they decide is right is economic slavery. As an initial metric, the number of hours the average family must work in a lifetime to gain a reasonable living is a working measure of prosperity. In my grandfather's generation, only the fathers worked and raised large families and mothers were expected to maintain the family. In my father's generation, fathers were expected to work for a career and mothers were welcome to work and to supplement the family income. In my generation, many families had to have two income earners to be effective economic units, hiring others to care for younger family members so that both parents could work. Our long-term witness shows a declining prosperity. We have a working rule for this from performance engineering. If things are failing in spite of what you are doing and you continue doing the same things, the result will be future failures. If you continue doing the same things, but with greater effort and intensity, it will yield an even more spectacular failure. Our economic performance increase will require fundamental changes in direction. We have attempted changes in our economy in the past but still had a resulting loss of prosperity. Doing these same type of changes more intensively invites more spectacular declines. That is the reality of performance. In modern commerce, we have business largely performed under corporate leadership with external management through legislation and government regulators. Our economic efforts are through corporate business entities run by administrative management leaders. Each business performs the economic function. It consumes the time and effort of employees to produce goods and services. When the businesses are most effective at this, the economy will be most effective at satisfying public purposes for having an economy. We have the immediate challenge of corporate leaders who firmly believe their management decisions will determine the success or failure of their businesses. The only real decision makers who determine business income are the customers who buy or refuse business products. For other corporate relations with outsiders, the corporate owners are just the source of a corporate capital resources with the corporate leadership deciding how much of the business earnings they should distribute to the investors. These leaders are corporate employees making decisions on behalf of investors with neither effective agency established nor personal responsibility accepted as to those investors. To challenge this further, the administrative leadership of modern corporation sees itself in ongoing competition with labor, the working employees. Put this into our understanding of the purpose for an economy, the people's purpose. The modern corporation is not focused on representing the owners except on a trust me basis that assumes support for internal corporate decisions. These leaders do not intentionally serve their customer base but try to align their operating decisions with what the public is willing to buy. They do not intentionally serve their employees, the working public, but call upon the working public to support whatever leadership decides upon as a corporate direction or effort. Our major corporate business units have been witnessing the result in terms of increasingly exporting their business efforts outside this nation so that they no longer even serve the working public. These corporate businesses are legal constructs that should only exist for the purposes of we the people. The legislation that continues to support their existence like the laws itself has been provided based on sovereignty of leadership on a business aristocracy that gets its living through the productivity of the people working within the corporation. What we have been doing creates internal division between the leadership and labor, though both are hired to operate the same business. As noted in earlier lessons, there is no less effective way to do anything than to divide and do competing camps where what one side can accomplish is limited to what the other side cannot prevent. The technical word for performance engineering, the technical word from performance engineering is waste. Waste is cost without benefit. Type two waste is output from the business that neither earns nor contributes to earnings. Type one waste is incoming resources that do not contribute to business performance. Taxing businesses is waste. It is removal of income from investors in terms of what no longer comes to them as return on their investment. It is removal of income from employees in terms of what business cannot earn to pay them the income from or benefits. It is removal of income from customers in terms of what is taxed business must charge to its in its products in order to stay in operation. A tax on business is just another way for a sovereign ruler to seize on the income of its people. The business purpose for English common law was support for the commerce of the common people. It was as a farmer who raised cows so he could sell their milk. It was seeing to the economic welfare of common people so the aristocracy could prosper off of what they earned by their productive efforts. This basic purpose has not changed through our American revolution but the aristocracy was denied the ability to tap into the common to what the common people earned. It was having a government protect and promote commerce without having to support an aristocracy. Taxation has largely built a new American aristocracy in government. One that harvests the commerce of the people to support what leaders decide is best for people. Government efforts have arranged an economic aristocracy largely through laws of incorporation. The welfare of the people may well still be the purpose of individual leaders but the modern business legal system is otherwise directed. Regulatory business management is a way to deny the sovereignty of citizens. Where the government draws taxes from unwanted business efforts it draws taxes from the people. Where government pays public dollars to lower prices. This is activity interfering with economic freedoms of many through expending the prosperity of everyone. Economic regulation provides no valid economic purpose for expending public dollars. If the people will not support a regulatory purpose on their own having the government tax the people to expend on it fails the test of representing the people. Public spending on publicly determined actions has been ramped up until it is a severe blight on personal prosperity. A tax bill exceeding 25 percent of all that a person earns severely damages prosperity of citizens. Wherever government spending supports a huge part of the public they are not producing value for other people to receive. Non-productive people are a drag on the economy. The productive capacity of the United States is awesome. It has been able to keep us from utter poverty for decades even with sovereign leaning governments sucking prosperity out of it. There are indeed changes to be made by following generations. They will not be effective through additional government spending but through redirecting government to actually serve the public supporting the economy by insisting that our business efforts deliver what people value to the people who value it. The change will not be through government creating employment through spending public dollars but through working to eliminate the waste that is sapping our economic performance. The change will not be through government regulation of industry but through empowering citizens to own and operate corporate entities keeping our business entities focused on serving their owners their customers and their employees. The change will not be through more of the changes that have been done in the past supporting the sovereignty and independence of business aristocracy but through serving the sovereign citizens who invest who work and who buy goods and services to meet citizen wants and needs. Instead of government trying to serve the people through regulatory industry it will be less expensive and more productive to empower investors to set management upon corporate operations empowering customers to assure that they receive what citizens value and empowering employees to assure a fair day's wages for a fair day's productive efforts. The direction of change is simple enough it is serving the people instead of serving the economic aristocracy it is performing the function of supporting the people instead of first creating artificial people and then selectively supporting those created beings. With this we have that all important legal purpose for government management over our operating economy it is to empower the people to own and operate their economy supporting the people as they arrange for commerce to serve their needs and wants. It is to protect their productive efforts from interference and support their efforts at coming together in corporate efforts to operate businesses that serve the people. It is to support the productive commitment of people as a way to earn a living. It is to promote the effectiveness of citizens through education and training giving special support where it can enhance the effectiveness of citizen efforts.