Adam Pohl here with Toddy Korbel and Toddy. Boy, you guys put forth such an outstanding effort going against a 4-0 team. And the first thing that stood out when you looked at the stats of this game is that Catholic had scored 15 goals in their first four games and you're able to hold them scoreless in 110 minutes tonight in this scoreless draw. Yeah, I thought it was just an outstanding effort, really team defense across the board from, you know, from our forwards through our mids and certainly to our back line. And I thought that our backs of Gabby Arters and Allison Arters and Alex Hamill and Riley Patterson and Keegan and Goal just really limited Catholic's chances. I thought they did a really good job keeping them in front.  They're averaging four goals a game roughly. And we were very, very organized and I thought they just did an outstanding job. Toddy, it seemed to me that in the first half you guys really kind of took it to them and almost deserved a better result than being scoreless at the break. You know, that's been a little bit our entire season so far. We're 04 and 1. We've played some, you know, some of the best teams in the country. We started out, you know, with a really tough schedule. And I think we have one game that we, you know, feels that we'd really like to get back in our loss to, honestly, to St. Mary's.  We're really happy with how we're playing. And our record is not at all indicative of where we are.  And I think these things are aligning us, you know, to play our conference games. So I think many of these games could have gone either way. The overtime loss to Gettysburg, this tie, both against very, very good teams. So we're actually, you know, despite our record, we're really, really happy with where we are. And Toddy, lastly, you guys are at home. That's another thing. I mean, you started playing at neutral sites, a lot of road games. What does it mean to be back here playing here at Stevenson and having the opportunity to kind of get things rolling here because you're about to move into a conference playing just about two weeks time?  And, you know, some coaches, you know, sometimes it's good to get on the road because you get them really focused and you have them kind of trapped on a bus.  But I thought, you know, they've just, they've done a really good job adapting to, you know, rolling out of class straight into games and rolling out of class onto a bus.  And we've just, I think, across the board done a really good job showing up and adapting and, you know, bringing our game at every single game. Well, Toddy, thank you so much and congrats on a great effort tonight. Thank you.


Good morning traders, Patrick Muller here with another Tickmail chart that wants to draw your attention and say to developing opportunities here at the Eurodollar day time frame. In the corrective phase it looks like we are about to potentially complete an ABC equality objective down at 118.53 versus the 122.47 high. We also have the 61.8% retracement as the last swing to the upside at 118.85. What I'm watching for will be bullish reversal patterns in this 118.50 to <IP_ADDRESS> being the monthly projected range support set long positions, initially targeting a move back up to test 120 from below and that should then potentially be the first leg of a potential three-way correction to get a test of a monthly pivot from below at 120.85. So as traders plan the trade, trade the plan and most importantly manage your risk. Until next time, thanks very much.


Hey math learners, Mr. Marks here, your friendly neighborhood math teacher. Remember, it's not just about getting the right answers, it's also about learning and growing along the way. Now today's multiple choice problem comes to us from Pennsylvania's Keystone exam sampler pack and was user requested by at me as a toilet on YouTube. Now this problem involves a little bit of linear equations, so what do you say we uncover the mysteries this problem holds together? First let's read the problem's instructions. The equation 3x plus y equals 8 describes a function of x, which graph represents the function? All right, it looks like we have four graphs to choose from here. There are so many ways that we could proceed, it could be quite overwhelming. What ways are you thinking of for today? What do you say we first start by making some observations? For each of them, let's first note the y-intercept. The y-intercept of a line is the value of y, where the line crosses the y-axis. So for a, our y-intercept is 8. For b, our y-intercept is 0. For c, our y-intercept is 8. And for d, our y-intercept is 8. Hey, what a great start. Next let's make some observations of the line's slopes. Remember that the slope of a line at its plainest is simply a measure of a line's steepness moving left to right. So for a, our slope is negative. Because from left to right, the line appears to be downhill. For b, our slope is positive. Because from left to right, the line appears to be uphill. For c, our slope is positive. And for d, our slope is negative. Fantastic. We have quite the collection of observations at this point. Now it's just time to read the problem's instructions one more time to see how we might use these observations to our advantage. Now we are being asked to find which of these four graphs represents the function 3x plus y equals 8. Currently, this equation is written in standard form. Or ax plus by equals c. But because we have these great slope and y-intercept observations, what do you say we rewrite this equation in slope-intercept form? Or y equals mx plus b, where m is the value of the slope of the line and b is the value of the y-intercept of the line. First, we need to move the quantity with the x variable. So we will subtract 3x from both sides. And with this simple change, we now have our equation written in slope-intercept form. So the y-intercept that we are looking for has a value of 8. And the slope that we are looking for has a value of negative 3. All right, now let's use this information to narrow down our choices. Option A has a y-intercept of 8 and a slope that is negative. So it could still be A. And look, option B has a y-intercept that is 0, which is not equal to 8. So we know it can't be option B. Let's cross that option out. Hey, an option C has a slope that is positive, which could never equal negative 3. So we know it can't be option C as well. And option D has a y-intercept that is 8 and a slope that is negative. So that leaves us with options A or D. Just like that, we've narrowed down our choices. But where do we go from here? How do we decide between option A and option D? Again, there are so many ways that we can proceed. It can be quite overwhelming. What ways are you thinking of? For today, let's determine which of these two remaining options has a slope of negative 3. And for that, we're going to use our handy-dandy slope triangle and the concept of rise over run. First, we'll mark and note two points on option A's line that have easily named coordinates. Mark one point here, not here, not here, oh, and here. Next, draw horizontal and vertical lines from the points to form a right triangle. Count the vertical distance to determine the rise. Here, we go down one, so the rise is negative 1. Count the horizontal distance to determine the run. Here, we go right 3, so the run is 3. Now slope is rise over run, so option A's graph has a slope of negative 1 over 3, which is not equal to negative 3. So we know it can't be option A. But wait, there's more. Though we now know the answer is D. I want you to confirm whether D really is the answer or do we need to rethink our steps. Can you calculate the slope of option D using rise over run and confirm that it's negative 3? Take this as an opportunity to challenge yourself. Additionally, why don't you ponder these questions? What error in thinking would lead somebody to select A? What about B or C? Let me know what you come up with. Hey, props to you for taking some time out of your day to do some math with me. I hope you followed along, and if you made mistakes, that's all good. Just remember that every mistake is a step towards learning something new. This is Mr. Mark, signing off. I'll see you next time with another math problem. What did you think? Did you approach this problem differently? Let me know in the comments. And if you enjoyed this problem, show your support by liking and sharing this video. And don't forget to follow my page to stay up to date on more math related content. Until next time.


Welcome to our Open Talk series. My name is Teri Zatzuga and I'm the research lead of the AI and Society Lab. I'm leading an interdisciplinary research group that wants to find out how AI can serve the public interest. And pretty early in our research we understood that this is such a big and complex question that we cannot solve it completely by ourselves. So in this series of conversations we speak to people who bring in their experiences and their research to find out the limits and the potential of AI to serve the public interest. And we really hope you enjoy these conversations. My name is Stefan Dreier. I'm one of the senior researchers, the one for media law and media governance at the Leibniz Institute for Media Research. And I've been working in this position for almost 20 years now. So I'm a lawyer by training and my research focuses on general transformation of media use, media power and the regulatory challenges that brings with it. Now I'm speaking to you as one of the PIs of the project financed by the Bavarian Institute for Digital Transformation. That project is called Coding Public Value. The main context and the problem that we want to clarify a bit is that in changing media landscapes there is ongoing pressure and public debate on how public service media companies or broadcasters can change with these developments. So one sentence regarding this public service media companies, those are the ones established by law in Germany and that have a specific public value remit. The idea here is that more and more media offerings in online environments and the question is if you use offerings in online environments you need software to do so and how these very general and broad remits can be applied to those software environments and especially to the engineering process of those software products. As you might already know this is not only a legal issue in the sense of regulatory requirements engineering but they are also deeply involved into public discourse and political discourse regarding the expectations that society has towards those broadcasters. The software engineering part plays a specific role within this project because we want to come up with ideas and options on how to translate those expectations into formalized software. How would you define public value? We try to come up with a definition of public value and public value is not even a vague legal term that you can find anywhere in legal texts. Public value is as itself an estimation or an expectation towards specific organizational structures and forms of conduct that might lead to public value as an outcome. You search for requirements necessary to the engineering process to achieve public value. Does this change the process of engineering generally? The answer is yes. This changes to a significant degree how you develop software because this is not only user or business model focused but you have to consider more public or stakeholder related requirements as well. This is kind of a deliberation. The deliberative part is already proceduralized in public service media. So they have internal boards, for instance, that check on how those concretization has taken place and whether this is still within the public value remit. There is no single true understanding of democracy. How do you translate this diversity into a proper media diversity? There's also a part of service offerings where I as a user might be able to decide on what algorithm I want to select for my content. And this is where we can apply the concept of diversity, not on the diversity of content but on the diversity of selection mechanisms that I can choose from. So this is very much a level of understanding of diversity that can now be applied to software of those public service media providers. So that's a very interesting thing to have because then we have to think about user interfaces, for instance. What does that mean? How can I explain to an end user that there are different understandings of democracy understanding based selection models for diversity or diverse content? What's your opinion on the challenge of translating requirements from the regulatory sphere into software development? We have discussions about this challenge almost every week between us and the software engineering colleagues. There's so much going away in translation already between lawyers and software engineers. If you look at the latest proposal on the AI regulation of the European Commission, it was I was shivering because there's so much implicitly going on into those provisions. So this is the first challenge coping with those legal terms that somehow should be applied or have to be applied into software without understanding from a legal perspective how software engineering works. And the second challenge that we saw is that legal understandings of words or norms change over time because of court rulings, for instance. And that is again not how software is being built. Once the system has gone live, you will optimize it towards a better user experience. But you are seldom using the software to get a feedback loop to the legal requirements. So we need a model that is also available in life systems to be sustainable or sustainably compliant with the legal requirements. And for that, you need someone who is also monitoring changes on the legal level. Our open talks, our open for collaboration, contact us to get involved.


Hello everybody. My name is Aditya Kulkarni and I'm a senior research scientist at Lantern Pharma. I would like to thank the Oncotar Get Journal and its associated publishing house for the opportunity to present our cancer drug development research activities on this platform. I've been a biopharmaceutical scientist, manager and inventor combining over 12 years of research experience with a vision to direct drug candidates from the bench to the bedside. I'm a biochemist by training having applied it in the fields of cancer biology and small molecule drug development. One of my areas of interest involve identification and validation of biomarkers of drug response and resistance and at Lantern Pharma I've been fortunate to have explored and contributed to several areas of drug development. Lantern Pharma is a clinical stage oncology focused by a pharma company leveraging artificial intelligence methodologies to generate gene signatures predictive of response in patients to its pipeline of genomically targeted therapeutics. Using this precision medicine approach Lantern Pharma aims to potentially deliver best in class outcomes using the method of targeting drugs to patients whose genomic profile is going to identify them as having the highest probability of benefiting from a given drug. Now Lantern Pharma is having four drug candidates in development all in various cancer types having unique and unmet clinical needs. One of these is LP184 which has a very interesting history as well as a unique anti-cancer activity profile which brings me to the topic of our publication. The title of the paper which I am going to be describing in today's session is LP184 retains nanomolar potency in non-small cell lung cancer carrying otherwise therapy refractory mutations and as the lead author on this study I'll make sure to break down the title for everybody to appreciate. Our paper is essentially describing the activity of LP184 Lantern Pharma's lead preclinical drug candidate in a wide range of preclinical models of lung cancer specifically non-small cell lung cancer which accounts for 85 to 90 percent of the total lung cancer incidence. LP184 belongs to a class of compounds called acylphalvein agents which further can be categorized as DNA damaging agents. In this paper we also identify genetic correlates and determinants of LP184 sensitivity that will facilitate further biomarker development during during clinical translation and allow additional personalization of therapeutic options for future treatment of non-small cell lung cancer. With this I'd like to talk about a bit of a history of the research leading up to this paper. Our company name Lantern Pharma actually comes from the lantern mushroom that naturally produces a fungal toxin called eludin. The first identified examples of eludins were isolated from the lantern mushroom at the New York Botanical Garden in the 1950s. Eludins were then extensively studied for their cytotoxicity in various tumor types in cell lines as well as xenograft models. However frequent animal deaths associated with high eludin toxicity restricted their use as potential anti-cancer agents. In an effort to obtain cytotoxic compounds with favorable anti-cancer properties, in an effort to obtain compounds with favorable anti-cancer properties, acylphalveins were derived as semi-synthetic analogs from eludins. And this class of compounds with the core structure of acylphalveins and their analogs were much milder cytotoxins than their natural precursors eludins and exhibited favorable tumor specificities. Now I want to point out four highlights that emerged from this early generation of acylphalvein molecules. Number one is the retention of sensitivity of acylphalvein molecules in multi-drug resistant cancer cells that were otherwise resistant to commonly used chemotherapy agents such as cisplatin, doxorubicin and ironotecan. The second highlight is that acylphalveins are uniquely tumor targeting while sparing normal or non-tumor cells. The third highlight is that acylphalveins show heightened activity in cancer cells with compromised DNA repair capacities. And the fourth highlight which is the most remarkable is that acylphalveins are pro-drugs that are activated by prostaglandin reductase I or PTGR I which is an oxidoreductase enzyme and has now been established as a fundamental driver of tumor sensitivity and appears to be a unique and stringent efficacy biomarker for this class of compounds. High level of interest in this profile of acylphalvein bioactivity and mode of action further triggered the development of new synthetic strategies to produce newer generations of acylphalveins and their analogs. We identified LP184 as a next generation acylphalvein analog with highly improved and favorable therapeutic window potency and a validated role of PTGR I in its cytotoxicity along with preliminary hints of synthetic lethal relationship with DNA damage repair defects. It isn't surprising that certain mutations in oncogenes and tumor suppressor genes underlie non-small cell lung cancer likely by acting as drivers of tumor genesis or metastasis or therapy resistance at some point in the disease and treatment progression. But what is surprising is that if we consider just four genes, two oncogenes KRAS and KEEP1 and two tumor suppressor genes CP53 and STIC11, these combined account for more than 40% of non-small cell lung cancer cases. And it is not surprising that alterations in tumor suppressors or oncogenes underlie non-small cell lung cancer likely by driving tumor genesis, metastasis or treatment resistance at some point during the disease and treatment progression. What is surprising is that if we consider a handful of these tumor suppressor or oncogenes, namely KEEP1 and KRAS that are famous oncogenes and TP53 and STK11 that are tumor suppressors, these four genes account for more than 40% of non-small cell lung cancer cases. And there are hardly any therapies that are effective in patients having mutations in these genes. Furthermore, it's important to note that the signaling pathways and interaction networks in which these genes operate are non-overlapping or non-redundant and hence it's very difficult to develop any single class of agents to target non-small cell lung cancer having alterations in these genes. And we believe that LP184 could really have a big impact potentially in these molecular subsets that appear to be very segregated. I wish to talk about the most notable features of our work published in this paper and that can be summarized in six points. One is that LP184 is effective in vitro, not only in 2D models and in primary non-small cell lung cancer models, but also in 3D models of metastatic lung cancer models, especially brain metastasis originating from primary non-small cell lung cancer. And this is especially relevant for application in brain metastasis since we also found that LP184 crosses the blood-brain barrier in an in vitro model of multicellular human blood-brain barrier. The second feature of our work is that LP184 sensitivity correlates positively with transcript levels of PTGR1 in the panel of cell lines tested. This supports the hypothesis that PTGR1 is very important for the cytotoxicity of this compound. The third highlight from our work is that LP184 activity actually turned out to be independent of mutations in the four genes that I mentioned previously, K1, KRAS, STK11 and TP53. When we compared cell line subgroups with and without mutations in these genes, there was no statistically significant difference in the IC50s across the cell lines. LP184 retained in vitro activity even in the presence of deleterious mutations in these genes that would otherwise be associated with chemotherapy resistance in non-small cell lung cancer. The fourth highlight from our work is that LP184 turned out to be up to 3,000 times more potent than commonly prescribed chemotherapy agents such as cisplatin, toxorubicin, genocytobine and oxalipatin. The fifth highlight from our work was that LP184 dose ranges that were safe and tolerable were identified in vivo in the mouse model along with demonstration of anti-tumor efficacy in the xenograft mouse model of non-small cell lung cancer. And the sixth highlight from our work, which is the most clinically relevant, is that elevated PTGR1 co-occurs with mutated KEEP1 in clinical samples upon retrospective analysis from clinical databases. Non-small cell lung cancer especially is one of the very few cancer types in which the response biomarker PTGR1 that underlies LP184 activation actually correlates very significantly with the molecular drivers of the disease that are often mutated and even undruggable, for example KEEP1. Here we see a clear overlap between the drug activity and the area of unmet need in non-small cell lung cancer. And this also makes sense mechanistically considering the regulatory relationships between components of the KEEP1-NRF2-PTGR1 axis. When we have wild type KEEP1 in cancer cells, represses NRF2 and does not induce PTGR1 above basal levels. However when KEEP1 is mutated as seen in up to 20% of lung cancer patients, it does not repress NRF2 which is then free to translocate to the nucleus and induce its target genes including PTGR1. So there you see how KEEP1 mutation could really drive the upregulation of PTGR1 which in turn would sensitize the tumor cell to LP184. Now I'd like to talk a bit about what we would like to do next, what's in progress and where we would like to see our research being taken. We find that LP184 is a promising small molecule with activity in multiple solid tumor types. Here we continue to identify patient derived models that correspond to the clinical and molecular niches that LP184 is predicted to be effective in. We'd really like to see translation of our cell line based in vitro and in vivo work to a demonstration of efficacy and safety in patient derived models be it ex vivo 3D models, fresh tumor biopsies or xenografts not only in non-small cell lung cancer but in other solid tumor types where strong rationale exists to pursue the development of LP184. Predominantly undruggable non-small cell lung cancer is associated with either KEEP1 or KRAS mutations that account for 25% of lung cancer patients and there are very few therapeutic options that are only recently emerging to address these patients. Further, there are also uncharacterized or non-detectable mutations or alterations occurring in up to 35% of non-small cell lung cancer patients. Now when we compared overall survival and time-to-treatment failure within two subgroups of patients, KEEP1 mutated and KEEP1 wild type receiving the same standard platinum doublet chemotherapy, the KEEP1 mutant patients perform much worse. Their outcomes are much inferior to the KEEP1 wild type patients. Also KEEP1 mutant patients are more likely to undergo metastasis. So we'd like to really propose LP184 as a treatment option in three different areas. One as an option for KEEP1 mutated lung cancer, two as an alternative to cisplatin in patients that are ineligible to receive platinum based therapies and three as a combination with radiation therapy as part of the palliative care for late-stage cancer patients. There are other areas also that we envision LP184 entering the clinic and those include DNA damage repaired efficient solid tumors such as pancreatic cancers having defects in the homologous recombination pathway or the nucleotide excision repair pathway. Furthermore, since we have preliminary hints that LP184 crosses the blood-brain barrier and shows in vivo antitumor effects in aggressive brain tumors in mouse models, it also becomes a promising candidate for CNS tumors as well as brain metastasis. And at the end I would like to extend acknowledgments to the entire Lantern team of my colleagues, supervisors and executives as well as our partners at ReproCell without whose support this work would not have been possible. Thank you.


So, there are, as I mentioned to you a little earlier, there are two ways of looking at matrices. The first is that it is a rectangular array of scalars. That is a simple way to introduce matrices, that is why I started with that definition. But as I mentioned, the more useful viewpoint is that it represents a linear transformation between two vector spaces. So, in order to understand that, we need to know what a vector space is and that is what I am going to define next. Are there any questions so far? Sir, you said we should perceive matrix multiplication as a linear transform. So, I mean, in no dimension, it is very easy to say that when we do a linear transformation, what happens? But in higher dimension, I mean, how should we ensure that this is a linear transform? Can you repeat your question, please? Sir, I was saying that, you said that, I mean, matrix multiplication we can perceive as a linear transformation. Yes. So, I was asking that, how can we visualize it? Like in lower dimension, it is very easy to see that this is a linear transform. But in higher dimension, let us say going from one to one, how can we see? Yeah. So, it turns out, I mean that we cannot visualize more than three dimensions. So, if it's two dimensions or three dimensions, I can kind of draw things or I can show you what happens in three dimensions and so you can visualize it. But there is no hope of visualizing a linear transform from, say, six dimensional space to eight dimensional space or 16 dimensional space down to 14 dimensional space and things like that. You cannot visualize it. So, it's a mathematical construction and you have to take it as such. But it is, that is what it is doing. It is taking a vector from 14 dimensional space and then mapping it to, say, 23 dimensional space, something like that. So, that's what it is doing. You cannot visualize it. So, I wanted to understand, I mean, what does it distinguish between the linear transform and the non-linear transform? So, how can we distinguish between these two transforms? Okay. So, this refers to how do we define linearity? Okay. So, I'll come to that in a little bit. For that, I need to understand this concept of vector spaces and how we define a linear transform between vector spaces. Okay. So, you do need to understand, we have to cover vector spaces before I can formally define what a linear transform is. But for now, I'm just saying that there are two ways to visualize or view matrices. One is a rectangular array of scalars. The other is that a matrix represents a linear transform between a pair of vector spaces. And the key point is that any linear transform, so I need to define vector spaces. So, there is an object which is called a vector space. And if I define two vector spaces and if I define a linear transform between these two vector spaces, that can be represented as one and only one matrix. So, there's a unique mapping or a one-to-one mapping between linear transformations between two vector spaces and the space of matrices. So, we'll come to that shortly. Okay, so let's start with vector spaces. So, in order to define a vector space, we have to start with a field. A field is a set of scalars and for the purpose of this course, we are only going to essentially focus on the real or complex field. So, that is the set of all real numbers or the set of all complex numbers. So, in the back of your mind, even though I write F here, think of it as a shorthand notation to say it's either real or it's complex. It's a set of scalars with two operations defined on it, plus and dot. And it's closed under plus and dot. That is, you take any two scalars and add them together. You will get another scalar which belongs to this field F. And you take any two scalars and multiply them together. That's this dot symbol. Then you will get another element that belongs to this field F. Both plus and dot are associative and commutative. There exists an identity element, both for addition and multiplication. And every element has an additive inverse. So, given any A belonging to F, there is a minus A which also belongs to F. And all elements except the additive identity, which is typically denoted by 0, have a multiplicative inverse and multiplication is distributive over addition. Again, this is a very formal sounding definition. But like I said, for the purposes of this course, just keep in mind that we are thinking about the real line or the complex plane and the usual multiplication defined in the real line or the complex plane or multiplication of complex numbers. So, there's nothing new here, but there is a formal way to define these things. I'm not going to deal with these too much, but I put these down mainly for the sake of completeness so that you know where these things come from. Now, a vector space. A vector space, I'm going to use either S or capital S or capital D, or capital V to denote a vector space. It is defined over a field F and it satisfies two core properties. If I take X and Y belonging to this vector space S, then their sum X plus Y also belongs to this vector space S. And this sum is defined as element wise addition. If I take X belonging to this vector space S and any C belonging to this field F, then C times X also belongs to S. And the scalar multiplication is defined as multiplying every entry of this X. And these elements of S are called vectors. And addition and multiplication, which are used here, satisfy some set of eight actions, which I'm not going to list here. But again, for the purposes of this course, let's just think of it as element wise addition. And multiplying every entry of this vector X by this scalar. So this X plus Y as defined here is actually taking a simple linear combination of these two vectors X and Y. A more general linear combination, if you are given vectors v1 through vn in each vi belonging to r to the m, this is the m dimensional real space that is a set of vectors with m real valued entries in them. And i is 1 to n, those are the n set of vectors. And if you're given scalars i equal to 1 to nci, then if I define a vector y, which is equal to the summation i going from 1 to n, ci times vi, that is called a linear combination of these vectors v1 through vn. We also write it often by stacking these vectors v1 to vn as a matrix. Then this matrix will be of size m by n, because each of these vectors are m dimensional vectors. And we stack the entries of the elements of this ci as a column vector c1 through cn. So this is n by 1. And when you take this product of these, this matrix vector product as I defined earlier, then that's exactly the same as doing summation i equal to 1 to n ci times vi. The moment we define linear combinations, we can define linear independence. So a set of vectors v1 through vn are linearly independent when summation i equal to 1 to n ci vi equals 0, if and only if, c1 equals c2 equals etc equals cn equals 0. It's important to take a minute and digest this definition. Again, this is something you would have seen in your undergraduate course. But one important thing I want to point out here is the if and only if condition. The if part is trivial here. Of course, if c1 c2 up to cn are equal to 0, then summation ci vi is always going to be equal to 0. 0 times a vector is a 0 vector. And so when you add up all the 0 vectors, you will get another 0 vector. This is a 0 vector here. So the if part is trivial. So really the crux of this definition lies in the only if part. That is, there is no other linear combination of these vectors v1 to vn that you can take and obtain the 0 vector. So graphically, the way to think about it is if I have a vector v1 like this, another vector v2 like this, then can I take a linear combination, scale this by c1, scale this by c2, add them together and then end up at the origin, get the 0 comma 0 vector. If I can do that, then these two vectors are linearly independent. If not, they are linearly dependent. It turns out that these two vectors are linearly independent. And this is something that should be obvious to you. And instead, if I take three vectors like this, now it turns out that I can always find a non-trivial linear combination of these three vectors such that I will end up at the origin. So three vectors in the two dimensional plane are always going to be linearly dependent. And so we say that a set of vectors are linearly dependent if they are not linearly independent. Again, continuing with the theme of linear combinations, the span of a set of vectors v1 through vn is the set of all y's which can be written as linear combinations of these v1 to vn. It turns out that this is a vector space. And again, this is something that you can try to show. It is very easy to show this. The point is basically that if you take two vectors belonging to span of v1 to vn, then the first vector can be written as a linear combination of vi like this. And the second vector can also be written as a linear combination of these vectors. And therefore, their sum, so if it was, if the two vectors were y1 and y2, y1 plus y2 can be written as the sum of these vectors with different coefficients ci. And therefore, that also lies in span of v1 to vn. And similarly, if you take, if you scale a vector y by some alpha, then that is the same as scaling each of these coefficients by the same factor alpha. And therefore, alpha times y can also be written as a linear combination of these vectors. And it belongs to the span. So it satisfies the two properties we said that a vector space should satisfy. And so the span of a set of vectors is actually a vector space. A related object is the range space of a matrix A, which is the set of all y's which can be written as linear combinations of the columns of A. So y can be written as A times c for some c in r to the n. This is also a vector space. So essentially, the span of v1 to vn is the same as the range space of a matrix whose columns are v1 to vn. And the range space of a matrix is the same as the span of its columns. A subspace of a vector space is basically a subset of a vector space.  But it should satisfy a property that this subset of vectors is itself a vector space over the same field. When it does that, then we call it a subspace. So for example, if I take r2, then the set of all vectors y belonging to r2 such that y2 equals 0. That is the second entry of y is equal to 0. This is a subspace. Clearly, if I take two vectors whose second entry is 0 and I add them together, the second entry cannot suddenly become non-zero. And so that also belongs to this set. And if I take a y which belongs to the set and I scale it by some alpha, then the first entry will get scaled by alpha. But the second entry, which is 0, will remain equal to 0. So that will also lie in this subspace. We say that a set of vectors v1 to vn span a vector space S if the span of v1 to vn is equal to this vector space S. Sir, can you please once again elaborate on the subspace part? So a subspace is nothing but a subset of the vectors in a vector space. With the additional property that it should itself be a vector space. And a vector space is one which satisfies those two properties that I showed you earlier. The sum of two vectors in a vector space should be in the vector space. And scaling a vector by a scalar, you should continue to live in that vector space. You can never leave. Yes. Okay, I often joke if the physical class, I often joke that vector spaces are like Hotel California. You can never leave. Whatever you do, these vectors, however they interact with each other, you will always stay in that vector space. Okay, so if v1 to vn span a vector space, then span of v1 to vn is equal to the set S, this vector space S. In other words, any vector in this vector space can be written as a linear combination of v1 to vn. And any linear combination of v1 to vn is lying in this space. So this is another small point I want to make about, see the span of v1 to vn is a set of vectors and S is also a set of vectors. And when you want to say two sets are equal, that is equivalent to saying if I take any vector in S that belongs to span of v1 to vn and likewise if I take any vector which belongs to span of v1 to vn that lies in this set S. So they are equal. When this happens, we call v1 to vn as a spanning set. Okay, of course, it means like I said this equality means that every vector in S can be expressed as a linear combination of v1 to vn. Okay, so I think we have reached here and the next concept I want to discuss is that of a basis which we will do on Wednesday. Any more questions before we close the class? Sir, can you please explain this range space once again? The range space is the same as the span. The range space of a matrix A is the same as the span of the columns of A. And mathematically, it's defined like this. It's actually the same as this definition here. To say that y is in R to the m, where y can be written as a linear combination of vi is the same as saying that y is equal to A times C, where C is a vector in R to the n. It has n entries. It has C1 to Cn as its entries. Okay. Sir, is it equivalent to the column space of the matrix? Yes. So that's a good point. This is also called the column space. Sir? Yes. Sir, could you explain a span? So you take two vectors, okay, or in this case, as defined here, span is the set of all linear combinations of these vectors v1 to vn. So in other words, if I take just to, again, I'm trying to avoid going to one and two dimensions because like I said, linear algebra is not limited to one or two dimensions. But that's all I can show you here on a whiteboard. But if I take only one vector in two dimensional space and I ask what is its span, it's basically this line going through the origin. Okay, that's all the vectors that you can represent as a linear combination of this one vector here. But if I take two vectors in the two dimensional space, then their span is actually the whole plane. As long as these two vectors are linearly independent, by taking different linear combinations of this, I can span the entire two dimensional space. If I take two vectors in three dimensional space, and let's say I take one vector here and another vector here, then these two in this three dimensional space, but they will not span the entire three dimensional space. Okay, it's the set of all points that are reachable by taking a linear combination or taking the sum of scaled versions of the two vectors. Sir, could you repeat the plane part? Your voice was not audible for a preformant. Yeah, so all I was saying is that if I take, so if anybody is able to see the three dimensional axis that I've drawn, I've drawn the x, y, z axis, and you're able to see it, please confirm. Yes, x, y, z. Yeah, so if I take two vectors, one vector along the x axis, another vector along the y axis, you can see that if I take all possible linear combinations of these two vectors, I will span the two dimensional plane defined by the x and y axis. There will be now no component in the z direction, so it will span a two dimensional subspace of the three dimensional space. Okay, and that is true. Even if I take any two non-coincidental vectors in the x, y plane, together they will span the entire x, y plane, but they will never have any component along the z direction. Every vector I take, which is a scaled version of the first vector, will have a zero as its z component. So specifically, if I take v1 equal to, say, v11, v12, 0, and I take v2 equal to v21, v22, 0, it won't be 0, 1, and 1, 0. Any linear combination I take of these two vectors, c2, v2, will always be of the form v11 plus c2, v21, c1, v12 plus c2, v12, v22, and 0. So this third component will always be 0. So it will always lie in the x, y plane. Sir, we can't see what you're writing. It will come in a minute. Okay, so v1 and v2 span the x, y plane. So Vishnu has his hand raised. Yeah, Vishnu, go ahead. Sir, sir, earlier, can you hear me, sir? Yes, yes, please go ahead. The thing is, earlier, you said that vectors can be represented as columns of matrix, right? Columns of a matrix are vectors. Yeah, columns of a matrix are vectors, right? So is this like, is it compulsory to use columns or can you use rows also? Yes, so. But in lecture, it is mentioned as columns, mostly. Yeah, so that's where they say, there are three types of people in this world. The kind who think of vectors as column vectors. The kind who think of vectors as row vectors. Okay, so that was a bit of a joke, but essentially a vector, as stated, could be a column vector or a row vector. The point is one of its dimensions, it is a, when you say a vector, we are thinking of a one-dimensional vector. That is, it has one dimension, which is where you have, say, n elements. And it's a string of entries written along that dimension. You can represent it either as a row or as a column. And in fact, we'll use both depending on the convenience. But definitely from, it is true that vectors are often, it's most common to think of vectors as column vectors. Okay, thank you. Yeah, so in fact, if I go back here in my definition of matrix multiplication, I used both. I used a row vector and I used a column vector. Sir, what are the third kind of people? That's the joke. Sir, one more thing. Yes. Sir, like your, sir, matrix is like linear combination between two vector spaces, linear transformation, right? Yes. Sir, like, sir, we have matrices like m by m, right? m by m or something like that. m by m, okay. Yeah, m by n or something. But m by n, okay. m by m by n, like there are three, right? Tensors. Can. So I'm not discussing tensors just yet. Okay, that's okay. The matrix by definition in this course is going to be of two, there are going to be two parts to it, m by n. That's it. Okay. So I will not be at least in the, for the most of the part of the course, we will not have Okay, okay. by n by n. Yeah, thank you. I'll need another, another course to teach tensor mathematics. Okay. So if there are no further questions, we'll stop here. Thanks for attending. Sir, Ashu has a question. Yeah, go ahead, please. Hello, sir. In this last example, where you explained the two-dimensional vectors along the x and y. So here we took these two vectors along the x's, x and y, but if we take these two vectors along a certain plane, I mean one vector along xy plane and one vector along some other plane, say it xy or yz plane, then would it be able to span the three-dimensional space? What do you think? Sir, I guess we must have another vector to span three-dimensional space. Precisely. So that is one of the things that we will show, which is that you cannot span three-dimensional space using just two vectors, no matter how you choose those two vectors. Okay. If I take two three-dimensional vectors, okay, I can always find a vector in three-dimensional space, which cannot be reached as a linear combination of these two vectors. It makes intuitive sense, right? Because if I take the three-dimensional space like this, it's hard for me to draw it here, but if I take some vector like this, another vector pointing in some other direction, these two guys together define a plane. Okay. And it's only guys, only vectors that sit in this plane that I can reach by taking linear combinations. There's always going to be a perpendicular direction, which is 90 degrees to both these guys. And anything that sits in this 90 degree direction cannot be reached by taking linear combinations of these two vectors. Yes, sir. Okay, sir, thank you. Welcome. Okay, so I guess we'll stop here for today.


I think the concept of open source can be summed up simply by this statement. All of us are smarter than any one of us. The idea that people can from anywhere in the world participate in developing some of the greatest shared technology in the history of the world is the tenant of open source. Open source is really the coal and steel of the information society. Instead of being owned by the Carnegie's, it's owned by all of us. Examples of these open source projects are all around you. Linux is one example of a project that has created ten million dollars in combined value. It runs Facebook, Google, most of the internet, every major stock exchange in the world, every modern television, air traffic control systems, most of the world. It's really changed the way that technology works. Open source at its core is all about people. It's about the ability for anyone, wherever they are, who has access to the internet, some knowledge of computer science, and a willingness to share their ideas with others, to have others criticize those ideas in order to make whatever software they're creating better. That's an act of individual work that harmonizes with other individuals all over the world. The ability for anyone to kind of jump in and make a contribution and have somebody else who has knowledge about the software that everyone's working on to share that knowledge with another person. That, I think, is the ultimate educational tool. The ability for all of us to go solve these huge challenges together. And to be able to look at those problems that we've solved together and say, there's a little bit of me in there. I think a project like this has the same ability to be that way, to say, I did this with so many other people, but there's also a little bit of me in there. This XPRIZE is terrific in a number of ways. One, bringing literacy to millions of children throughout the world is in and of itself an incredibly worthy and huge goal. But what I love most about this contest is that the contest, in my mind, all of the work and interesting things that will happen here will begin when the contest ends. When all of the source code for this great learning tool is out there, and other people are going to be able to contribute to that and improve it and make it better, and a community will evolve around this, whose goal is to improve the tools that help kids learn to read. How great is that? A contest that will live on perpetually. That's what I think is the power of this open source approach that the XPRIZE is using with this effort. And at the Linux Foundation, we love it.