Hi, everyone. Thank you all for being here. I know it's too early in the morning, so thank you more for being here. So I want to start with a question.  If yes, I want you to raise your hand. Perfect. OK, I see you. I feel you. And I promise you, by the end of this talk, I will make my best so you have all the tools that needed to deal with any traffic drop coming your way. So allow me to introduce myself. I am Anna. I'm a circumsultant. I have seven years of experience in SEO. And I'm founder of Search Magic, an SEO beauty agency based in London. So let's give a context first. Google has a dedicated team working on their algorithm to improve the quality, overall experience, and relevance of their search results. So every three to four months, they release a big algorithm update. And once in a year, they release the core algorithm update. And it changes a lot the SEO landscape. Now, a brief of history. These are some of the most important Google updates happened over the past decade and more. We had Panda update, which was about identifying low quality websites. Then we had Penguin update, which was about identifying blackout signs of the link building. Then we had Medic update, which was emphasizing on the authority, expertise, and trustworthiness of pages that were affecting the well-being and happiness of a person. And then the most recent one, we had Helpful Content, which was about to identify if the pages have actually helpful content for the users. And a small note here is that, actually, we don't know what an update is when it's released. There is a lot discussion and research from SEOs all over the world to identify what's going on. And a fun fact here is that we give the names. So that's where you see all these weird names about Panda, Penguins, and all these animals. So a good point is, why should I care about an algorithm update? And I know you're right. There are three ways to get affected by an algorithm update. You might not affect it at all. You might affect it in a good way. Or you might negatively affect it. And that's the point that you care about, because you're actually losing rankings. So what this means? It means that simply, if you lose rankings, you get lower positions in search results for specific keywords. And when this happens, you get lower visibility, less people see you. And actually, you get less traffic in your website. And that, of course, means you get less sales. That's why you should care about. So if organic search is among your three to four top traffic sources of your website, it's definitely going to be an issue. So SEOs are really good guys. We're a bit weirdos, a little bit. We love devs. They make our life much more easier. But at the same time, we are really brave. You can't scare us very easily until we see in our feet this. Later today, we're releasing a broad algorithm update. Oh my god. It actually feels like this. It's like the clown from Eat in our living room saying, hi, I'm here for you. Oh my god. And I have to admit that even though I have worked in many traffic losses, every time I see this announcement, I'm like, OK, OK, here we go again. We're going to deal with this. So what we should do if we hit by an algorithm update, what can be done? And I'm afraid you're not going to like the answer. You can do nothing. And there is nothing you can do about it. If it's about to get hit, you will hit anyway. So at this point, you'll probably be like, OK, we wake up too early in the morning. Why are you even here? Why do you even do this talk? But I'm like, wait, guys. I need a moment to make my point. Give me just some minutes. So what a Google update, an algorithm update, actually means? It means that Google has rethinked the search intent of a specific page. And you're simply not relevant anymore, meaning that based on the user's data and user's behavior, they tend to search different kind of information of what they use to search. So yeah, simply as that, you're not relevant anymore. And if I want you to keep one thing from this presentation, is that Google is not like the big, punishing guys trying to find website owners that don't comply with their guidelines. They're more like the guys that they want to serve the best search experience. This is what they're working on. This is what their mission. And Google updates are simply part of the job. And to better understand this, I have a couple of examples. My favorite one is this one. It's about a huge publishing website based in UK. They lost over 70% of their traffic overnight. It was huge at this point. And this makes, actually, creates a lot of questions. But if we look closer to their page, this is actually one of their top ranking pages. They used to be top ranking pages. But if we look closer to it, we will see that in the same page, they're talking about politics, gossip, and fashion. So my question is, what is the point of this page? What is serving? What is the main target? Why should even exist? And more than that, why should even rank? And if we look even more closer, we will see that the Core Web Vitals scores, it's 11 out of 100, which means that probably a page takes 1,000 years to load, which, OK, leads, again, to the question, why this page should be in the top search results. And at this point, I want to tell that most of the pages, most of the category pages, were like that. So it makes totally sense why they lost their traffic. Another example, Cockapoo Food.  On April 2022, it was a keyword leading to a full informational search results. It was about research. It was about lists. But some months later, April 2022, it was leading to full transactional results. People, they just want to buy the Cockapoo Food. So that's why it makes sense to, you know, not be relevant. And OK, now we have a really good overview of what's going on and some more generic information about algorithm updates. But at this point, we need to see the next step, which is what I'm about to do with my manager or the stakeholders, that once they found out this drop, they're about to get mad. And this is not you.  This is definitely not a point of interest. If this is you, I'm jealous of you. I want your job. But for everyone else, we have a lot to talk about here. So first of all, we need to be transparent, meaning we have to address the issue very early on and make sure they know they are aware what's going on. Because if they found themselves, well, it's not going to be a good next step. And also make sure you have some solid next steps and actions to tell them so that will make them feel secure that you know what you're doing and you have it all under control. Next up, you need to manage expectations because the traffic laws might happen overnight, but it's not going to get solved overnight. It probably is about to get months to recover from this. So you need to make sure they are aware of it. And the last part is to educate them what happened, why it happened and that it's not your fault or the entire team's fault. It can happen, it can happen to anyone. And actually, it happens to a lot of websites. So, okay, how can we avoid get hit by an algorithm date? And to be honest, you can't avoid it at all. It's part of live, let's say. There is no specific checklist to go through and say, okay, I'm done with Google updates, but there is some side, some kind of factions you can take to eliminate the chances. So first of all, you need to be proactive, of course. So that means you have a really good SEO strategy in place that it's going to help you a lot. You can read Google's guideline updates to make sure you comply with them. This is also going to help you. And this is especially helpful if you have a new client. You can review their backlink profile. What we're doing here is to actually search if they have any signs of black hat strategies or they have a poor backlink profile. Also, we can check if there is any manual actions in the Google Search Console because all of these could be triggers for a traffic loss. And my favorite one is to create content across the buyer's journey in order to cover all search intent. So for the cocoa food example, what you can do is create content, not just the product page, create content for all stages of the funnel to make sure you are covered in case you lose some rankings. So for the cocoa food example, instead of just having the product page, you could also have reviews page, video content, guides, recipes, and any other kind of content formats. It's going to help. So we are done with the first part and we're going to the juicy part, which is how to deal with this traffic drop and what you can do to go over with it. And at this point I have another question. How you're about to deal with a Google, with a traffic drop or any algorithm update? I want you to raise your hand and let me know. Don't throw the laptop out of your window. It's not an option. I've tried it, it's not working. So don't try it at home. Okay, is there anyone who would like to speak and tell me how it's going to deal with a traffic drop? Okay, okay. Okay, let's go through the presentation then. Oh, yes. Any other ideas? I can see you, by the way, very well. So, okay, yes. Sorry? Updating, oh, updating, yes. Really good one, good idea. Okay, any other ideas? Anyone wants to say something? Okay, let's go through the presentation then. I created a spreadsheet that is going to help you to analyze any traffic drop, even if it's from an algorithm date or not. I was actually creating the spreadsheet during I was creating the presentation at the same time, so to make sure that I have detailed out all the next steps. So, the goal is to study the best period of traffic and rankings and compare it to the worst period of ranking and traffic. And to do that, we will need two tools, Google Analytics and Semras, or any other SEO tool that can provide historic ranking data, it's going to help. So for this example, I'm about to analyze an actual website with an actual big traffic loss, like really big. It's mightymoms.club and it has, again, a big loss out of 70% more loss. So, what I'm about to do first is to open Semras, add the website, go to the domain overview, add the website and go to the big traffic trend graph, and I'm about to spot the best period of time and I'm about to click this period. Getting a period of like a month, it's like a good amount of time. So, once I have the list, I'm about to get a full report of keywords rankings. I'm going to get a tons of keywords, especially if the website is really big. So, I want to do some filtering at this point to make sure that I don't go to analyze thousands of keywords manually, that wouldn't help. So, the filtering that I'm suggesting is a following. Keep only the top 20 keywords that they have a minimum search volume of 100, and also sort the list by estimated traffic. So, at this point, you will get all the best pages by traffic that they have a proper amount of searches. And now it's about to export the list. I'm going to get a thousands of metrics that I don't need, I don't care about them. The only metrics that I care about and I want to keep is the URL, of course, the keyword, the position, and the estimated traffic. So, I'm about to create a spreadsheet, create the seat one, which is SamRest data, and at this metrics, I'm going to call this spreadsheet SamRest data, and I'm keep forward. So, the next step, now that I have my SEO tools, let's say data, it's to look on the real data. And to do that, I need to cross-check my data with Google Analytics.  I'm going to find, again, the spot pick period of time. Hopefully, I've done correctly from SamRest, and it's the same period of time with SamRest. And if that's correct, good news. So, I'm going to click the period of time, again, a month, go to the landing pages report, sort by sessions, not users, and export the list. And for this example, the website that we were looking for, I want to tell you that 75% of the entire organic traffic was focused only in one page, a category page. And that was really bad. It put you in terrible position, very vulnerable position, mostly because if this page happened something to it, you're done. You have lost most of half of your traffic. So, again, we're going back to the Google Analytics list. Now, I have exported the list. Again, I'm going to get a ton of metrics and things, but I'm not need anything. I need only landing pages, sessions, and bounce rate. So, I'm about to create a second seat, Google Analytics data best period of time, and adding only the three columns that I need. And the tip here is from Google Analytics, you are not going to get like a full URL. So, what you need to do is to add this specific line that it allows you, it's like a formula, to get the full URL of the page. Okay, now, I need to cross-check what happens to the pages of the best period of time, because if it's a long time away, they might don't even exist anymore, and it's really important to know. So, I'm going over Screaming Frog, which is a really good auditing tool. I'm going to flip Screaming Frog to list mode. I'm going to add the URLs. I'm going to crawl only these URLs, take the status code, find out if they're, for example, redirected, either 404s and what's going on. And I'm about to create a third seat with the Screaming Frog data. So, for the example that we were talking so far, there was a really big amount of pages that were 404 pages, means they didn't exist anymore. And there was also many pages with non-available products that haven't been handled at all. So, that means actually that there were pages that make sense that they used to have traffic and they don't have traffic anymore. So, it wasn't even the Google update thing, it was just products don't exist. So, the print spreads, it looks like this now, and once you get the presentation, probably can be zoomed more. We have the Summers data, we have the Google Analytics data, we have the Screaming Frog data, and now it's time to combine all this data and make some conclusions. So, I'm going to need all the important metrics from both Semres and Google Analytics seats. And what I'm about to do is create a new seat, which is called Combined Data. I'm going to keep all the data from Semres, and I'm going to extract automatically the Semres data. In order to do that, I'm going to use a VLOOKUP formula. You don't have to go through and find VLOOKUP how it's working, I have it right there, exactly the formula that you need to use, and you're going to extract all the data for Google Analytics automatically with this way. But we're not done yet, we have more. The next thing is to check either any landing pages that exist in Google Analytics data. They have a big amount of traffic, but I didn't extract them so far from the Combined Data. So, I'm going back to the Google Analytics data, create a new column, again add the VLOOKUP formula to save time, and I'm going to extract the data and check if the URL exists or no. If the news are good, I'm totally caught up with both seats, so I don't need to do something else. And the last piece of this Excel is to get the current ranking position or the worst ranking position for these keywords that I'm trying to study. I have the best rankings, what they used to be, but I need to find out where they are today. And now we're going to the big finale, which means study the data. We're about to create a pivot table, this is going to help us a lot, identify the next steps, and what I want to do here is to find out for each URL what are the keywords and what is their ranking position trends. So, I'm going to use as main values the landing page and the keyword, and add some secondary values from traffic, position, and everything else. And I have right here exactly the steps that you need to take in order to create this pivot table and add the correct values. So, once I have the pivot table, I'm going to end up with a spreadsheet like this. So, for each keyword, not keyword, for each page, I have all the keywords, the rankings, and the search volume. And for each page I know exactly what are the keywords, which might be a big variety of them. For a page I might rank for like 20 keywords or more. What I want to find out is for each page what keywords had the traffic loss and what pages had the traffic loss. And if they have traffic or not, this is the important thing here. So, to help a little bit the situation, I'm about to sort out the page by organic traffic. So, at this case, I'm going to get the best pages in terms of traffic, and I can be able to see the keyword rankings. And that's it, actually. Then we have to find the patterns and study the data that I'm not going to do today because it might take a lot of hours. But let's recap the process. Actually, I'm about to find, first, the big period of time from Semras. Then I'm about to do some filtering because I don't want to go through thousands of keywords. I'm about to cross-check the big period with Google Analytics data. After that, I'm about to find the stats of the pages because they might not exist anymore. Create the combined data and put them all in Excel. Create the pivot table, and then it's time to identify the patterns and find the next steps. And at this point, I want you to remember we have two missions when a traffic drop is happening. The one is to defend the rankings that we have today and the second one to increase the potential and traffic of the rest of the pages. And here are some solid next steps that can work in any case. That would be to focus on pages that they have a big amount of traffic and they have a loss, probably, in rankings and work on their content and their metadata. Try to identify quick wins, meaning work on keywords from position four to position 30, and make quick optimization that's going to bring a really quick boost of traffic to your website. Also, sorry, focus on keywords that are, let's say, they don't have such a big amount of competition but they have a big amount of searches. And last but not least, create content across the buyer's journey and start creating more content that cover more search intent. And last but not least, try not to panic. I don't want to say don't panic because we're all humans. We are going to panic anyway. So probably I would finish this presentation with the next steps. I think it makes more sense. Just skip the next steps, the solid next steps. And thank you very much for your time today. Thank you. And yeah, we actually have time for some Q and A. So I'm sure you guys have a lot of questions but we have two microphones set up, one on each aisle. So if you have any questions, head over to the microphone right now. Depending on how many people there are, we'll just limit to one question at a time and we'll just be lingering over here. Hi, good job. Hi. Thank you for your presentation. Thank you. From your experience, is it better to create a 404 page for a page that does not exist anymore or redirect to the home page? I'm speaking if the page does not exist and there is no relative page to redirect the visitors. In my opinion, I would prefer to direct the page 301 not to hold a home page to the most relevant category or subcategory, it would make more sense. So you won't send older directions to the home page. Thank you. Hi, thank you very much for the presentation. Thank you. I just wanted to ask if a site got hit with any algorithm, will Google put the domain of that particular website into their bad books? Even though we are trying to match the level that Google is already asking us to do, like as you are mentioning, how will Google tackle that part? Will they just put us like they were the people whom we already hit on their faces? Let's put it that way. And now they are trying everything to get back, basically. Can you please repeat, because I think... No worries. Let's say that I have a URL like Malapras.com and we had 50% traffic coming from one page, which is services, for example. And we got hit by any algorithm update. Now, the URL will remain the same, Malapras.com. Okay, now as a company, we are taking the measure to get back over what we lost and everything. Will Google keep that URL in their bad books as actual URL, like even all the domain? How do they handle that? So you actually have focused the traffic to one page and you hit by an algorithm date, and this is gone, the page is gone. Okay, actually, this is what we said before a vulnerable position. What you can do is create more content around the topic and try to recover this from blog posts or articles or content. The domain authority also, then? Do you change domain or... No, domain will remain the same. Domain authority is not going to affect it by traffic loss, probably, unless you're about to lose a big amount of backlinks All right, thank you very much. Thanks a lot. Thank you. Hi, Anna, awesome presentation. Thank you. I would like to ask you from your experience how much time it takes to recover from a Google update. And what are the factors that influence this one? Well, it's going to take a lot of time, probably it's about to take more than two to three months, but it depends on how severe you will get heat on the algorithm update and what things you have to address, what was the actual problem. There were some slight cases that were never recovered from this, especially if you do nothing about it, of course you won't recover, but most of the times it's going to need three, four, maybe months, or maybe more, depending the case. Okay, thank you so much. Thank you. Hello, thank you for your presentation. So you were talking about a Google algorithm changes that were happening in the past, and it looks like that each of them had the specific intent or specific topic, how do you call it? So is Google like publishing or speaking about it or at least giving us a hint or a clue what they are working next so we can at least be a little bit prepared or know what is waiting us in the bigger known future? Actually, they do give us some kind of hints. They never announced 100% what's going on, but thank God the SEO community is really strong and they identify almost immediately what's going on, so that's the good news. I think after some months they are going to release more details after you found out, but the first months, most times they don't. So what would be the best place to check to find out about this? To find out what's going on with an algorithm date? Well, to be honest, I never actually checked one by one to find what is causing the algorithm update. What I'm trying to do is to solve the traffic laws. So probably they are looking, they study a variety of websites from different industries and they try to find common things that's going on. That would be a good starting point I think. Thank you so much. Thank you. All right, I have one as well. I just wanted to, no, I was just wondering if you could tell us about like the coolest or like the biggest traffic laws that you found and like how you fixed it afterwards. It was a directory website. It was one of the biggest websites in Europe. It was talking about professionals and professional works. They lost almost all their traffic, more than 80% laws. And actually we made up and create a different blog and some power pages to come up and rank the blog at least and redirect the traffic to the main website because things were a little bit bad at this point. Okay, we can. Thank you. Oh, there's another one, sorry. Yes. Hi Anna. Looking forward to the future because we've seen a demo at Google I.O. which is currently live in the US as in the Google Labs. That Google is showing more and more content on its own search result pages. I've heard numbers like 60% of all searches do not result in clicks anymore. What's your vision and strategy on this? I think we are going to see a thing called on Serps.io that at this point you will try to rank on the features of Google search results and get traffic from them. But we have to allow to see from the AI and all this. Yeah. Thank you.


In the previous video, I mentioned how a basis is a linearly independent spanning set, but I also mentioned without proof that a basis is a minimal spanning set. So if we have some, you know, subspace w, which is a span of so many vectors, right? So let's say that we like x1, x2, x3, up to maybe like some xp right here. And this is a spanning set, but this spanning set might not be independent. It could be that some of the spanners are actually linear combinations of the other spanners. And therefore we could throw them out without changing the span. Like maybe we give rid of x2, x3, and we keep all the rest. We could always prune down a spanning set. We could always prune it down to a minimal spanning set to make a basis. This means throwing out the vectors which are unnecessary. Now, how do we know which ones are unnecessary? Well, it turns out we're gonna look at a special case to that which actually provides the general method. Suppose we're given an M by N matrix A. And we've already seen already what we call two of the four fundamental spaces of a matrix A. So we've introduced the column space and we've introduced the null space. So remember the column space is the span of the column vectors of A. This will necessarily be a subspace of FM. And the null space is the solution set to the homogeneous system AX equals zero, which is necessarily gonna be FM. Now the dimension of the column space is what we call the rank of the matrix. Which remember, we define the rank as the number of pivot columns in a matrix. Well, I claim that the dimension of the column space will be the rank. That is the pivot columns of A form a basis for the column space of A. So the pivot columns are only columns you need in terms of spanning. And so if you have a set of vectors like we did before, right? So like I was mentioning right here, we have some set of vectors. Well, you can always put those vectors together. A set of vectors basically becomes a matrix. A matrix whose column vectors are exactly these vectors X1, X2, et cetera, right? So you put these vectors together in form of matrix. Then the span of those vectors will be equivalent to the column space of set matrix. As we row reduce the matrix, we identify who's a pivot and who's not. If we throw out the non-pivot positions and only retain the original pivot positions, that is we throw out the non-pivots and keep the pivots. The non-pivots in terms of the system of linear equations, they correspond to free variables. That means we have some freedom, but in particular we don't actually need them to span. And so if we throw out the non-pivot columns and keep the pivot columns, the pivot columns will correspond to an independent set of vectors which has the same span as before, AKA it's a basis. And so what one has to do if you have a spanning set, if you wanna prune it down, you just put those vectors together in a matrix, row reduce the matrix, identify the pivot columns, and then grab the original vectors that correspond to those pivot columns and you'll prune your spanning set down to a minimal spanning set, which will necessarily be a basis. I also wanna mention that the dimension of the null space is what we call the nullity before, which we actually defined it to be the number of non-pivots. So it turns out counting the free variables has to do with how big the solution set is for AX equals zero, but we'll talk about that a little bit more in another video. Let's focus on the column space right now and the rank, okay? So given this matrix A, which is a three by five matrix we can see right here, we're gonna construct a basis for the column space of A. We're gonna do this by row reducing the matrix. So we think of the first pivot position right there in the one, one spot. We're going to get rid of the numbers below the ones. So we're gonna take row two minus two times row one, and we're gonna take row three minus row one. So we get minus two, minus six, minus six, minus four, minus eight for the second row. That'll give us zero, one, zero, negative one, and positive one, like so. For row three minus row one, we get minus one, minus three, minus three, minus two, minus four. This will give us zero, negative one, zero, one, and negative one, like so. So then we move our pivot to the second spot right there since we finished the first column. I'm noticing that together this negative one here, I can just add row two to it. So we're gonna get plus one, minus one, and plus one. Oh, that gives me a row of zeros. This matrix now is in echelon form. We want echelon form because when we are in echelon form, we can identify the pivot positions. We do not need row reduced echelon form to do that. I mean, it's helpful, but we don't need it. We will now recognize that the first and second columns were pivots. So we will then select the first and the second columns of A. We go back to the original matrix A. We're not selecting the, we're not taking the columns of the echelon form. We just need the echelon form to identify where the pivots are located. We have to go back to the original matrix A and select its columns. Because if you take the span of these columns and the echelon form, that does not give you the same space as the column space of A, right? The span of these columns will be different to the span of these columns, but the location of the pivots do not change. The pivots were in positions one and position two. Therefore, the column space of A is going to be the span of, we take the first vector, one, two, one, and we take the second vector, three, seven, two. So I mean, by definition, the column space was the span of all five of these vectors, but it turns out, oh, we don't need the last three because those last three are combinations of the first two and therefore they just add extra redundancy to the system. We don't need them. In particular, the rank of our matrix A right here is two. There's only two pivots. The dimension of the column space is two dimensional and therefore we can take care of everything from there. Now, before continuing, I do want to kind of mention a metaphor, if we will, on what's going on here. In a previous lecture, I kind of made some examples about RC cars with joysticks and mentioned how that kind of has something to do with independence, has something to do with dimension, right? So I mentioned like, oh, my son has a car with two different joysticks on it. One joystick allows the car to go forward and backwards. The other one allows it to turn left and right. So therefore, by combining the two joysticks together, it allows my son to drive his car anywhere on a flat surface, right? It can drive anywhere on the floor. But let's imagine that his car was actually tricked out. He's got a couple extra joysticks on his remote, right? Maybe something like these extra joysticks for which the first joystick makes the car spin in a circle, right? And then the second joystick makes it drive diagonally, right? It'll drive diagonally based upon this. It's like, okay, you have these four joysticks now. Is the car able to travel more than two dimensions like the helicopter we had mentioned before? Like a helicopter can go forward, back, left and right, but it can also go up and down like a little drone. But the RC car, these extra joysticks are not ever causing it to leave two dimensions. These extra joysticks are somewhat redundant that if you want your car to spin in a circle, you can actually do that using forward, back, left and right.  This right here is just the sum of these two vectors. You don't need it. The circle one might be a little bit harder to convince yourself on, but it is still there. And so in terms of this discussion right here, if we have like this column space of four vectors, turns out two of the vectors were unnecessary. We can accomplish the same span with just two of the joysticks. And that's what the basis is all about. What is the minimal spanning set? What's a minimal set of spanners? That yes, I can get the whole span with five vectors, but I only need the first two. So these two are the only ones that are necessary. Well, why does the joysticks, why do these extra joysticks exist on the remote control? Well, it's, you know, sure, having more controls makes it easier to do tricks with the RC car. But if we're talking about just mobility, my son can drive his car to any place on the floor using just the two yellow joysticks, the two extra ones, although great for tricks, not necessary. Let's look at another example. And then we'll be able to do this one super quick, right? Honestly, the hardest part of these problems is row reducing the matrix. So if we have a matrix which is four by five, a four by five matrix right here, let's find a basis for the column space. We have to row reduce the matrix and you'll see that I already have the RREF computed for us, which you don't need RREF, any echelon form will do it. I just mentioned this one because if you have a graphing calculator, the RREF is a standard function. Just throw the matrix in there, RREF it, if I can use that as a verb. And there we go. We can then identify the pivot positions. There's a pivot in one, one, two, two. And then notice the last pivot is in the three, five position. This tells us that the basis for the column space will consist of the first column of A, the second column of A and the fifth column of A. Always take your, for the basis of the column space, you have to go back to the original matrix and grab its columns. So the column space of A, it'll be spanned by the vectors one, negative two, two and three, three, negative two, three and four. And then the last one here at negative nine, two, one and negative eight. So if you can get a echelon form of the matrix pretty quickly, then you can find a basis for the column space very quickly. Notice in this situation, the rank of the matrix is three. There are three pivot positions and you need exactly three vectors to span the entire column space. These other two vectors, although they might be great to have for doing tricks or something, they're not actually necessary in terms of the span. This column space is a three flat. It's not a five flat. We don't need those two extra column vectors inside of A.