Welcome everyone to our review for exam three for math 2270 linear algebra at Southern Utah University. If you're watching this video then it means you've probably made it to the end of the semester and so give yourself a pattern back and say whoopee and be excited that you're here. It's been a fun semester so far and let's not trip on the finish line and make sure we finish up this exam and do pretty well on that right. So I should point your attention to the instructions that exist for every exam. Make sure you do read through these and familiarize yourselves with the instructions and policies for this exam. They are the same as exams one and two but certainly things like the dates, the location, things like time limits. Sometimes these things do change from semester to semester so do make sure you check the information on canvas so you know exactly when the test is going to take place, where is it going to take place and how long do you have to complete the exam. Those things do change from semester to semester. This exam is going to contain 16 questions as you can see from this table. You will see seven questions coming from the multiple choice. They were with five points each. You will see three questions coming from the short response. Those are worth six points each and then there are six questions coming from the free response and they range from eight to seven points about the same. And then finally do remember to create and submit your no-card as that is worth two points on this exam. This test will cover the third arc of our textbook linear algebra done openly. This will contain chapter five about determinants, chapter six about eigenvalues and eigenvectors, and the vast majority of chapter four about orthogonality, particularly sections 4.3 up to the end of that chapter and then all of chapters five and chapter six, right? So if we were to write that down, so we're going to get section 4.3 to the end of the book, section 6.5 like that. On canvas, there is an exam syllabus that you can download that'll talk about a lot of these details as well. I recommend you do read through that as well. You can also find the link to the exam syllabus in the description of this YouTube video. And likewise, you can also find this document you see in front of you, this practice exam. It's available for download either through Canvas or you can download it directly from YouTube as well. So without further ado, let's jump immediately into the content here. Like we said, let's talk about our multiple choice questions. They're worth five points each. Please select what you believe to be the correct answer and only the correct answer. If there's no response, multiple responses are the incorrect responses selected. You don't get any points for these things. So make sure you select the correct response. And I don't really care how you do it. If you want to circle it, that's great. Checkmarks are great. Some people fill it in, whatever you're most comfortable with, just clearly indicate what the correct response is there on the paper. And there's only one selection that you should make. On question number one, we're going to see a question that involves the fundamental theorem of linear algebra, particularly if this is going to be placed on things like the nullity, the rank, the co rank, the co nullity of a matrix with some respect to these fundamental spaces like the row space, the null space, the column space, the left null space and their dimensions, right? And so if we're given some matrix, like in this example, we're given three, three by five matrix can and we're given some information about its row space, its column space, its left null space, or it's who would I forget column space, null space, row space, if we're gonna be information about those, what are the relationships we know? So the fundamental theorem of linear algebra tells us things like the rank and the co rank are equal to each other. And this is equal to the number of pivots that you'll find in the matrix in its echelon form. We know that the rank plus the nullity, that is always equal to the number of columns, aka n, as we often write this as an m by n matrix. And then we also know things like the co rank plus the co nullity is always equal to m, the number of rows in the matrix. And like I said, the rank and co rank are always equal to number of pivots. And so this tells us things like the nullity is the number of columns minus the number of pivots, the co nullity is the number of rows minus the number of pivots. So be able to compute these values given some of the information that's provided here. So like the rank nullity theorem as part of the fundamental theorem of linear algebra is what we would we want to study there, you can see more about this in section 4.6 inside of the textbook. Question number two is going to be a question about eigenvectors and that is recognizing you have an eigenvector. And so you'll be given a matrix a like you see right here, and you'll be given six candidates for vector, you can see it kind of slid off the page a little bit. There is another response over here. I hope that's okay if we don't show all the responses. And as always, I should mention as a reminder that the solutions to this test can be found in the same PDF document to scroll to the end of the document, you can see all the solutions in front of you. If you want to check your work and things. So can you identify what and whether you have an eigenvector for a matrix or not? Well, this one, you're going to be given six vectors and you're not asked to find the eigenvectors of just some arbitrary matrix, you're trying to find the are the eigenvalues of this matrix with these candidates. So at the very least what you could do is you could just take all six vectors and just multiply by a until you recognize you have an eigenvector or not, right? So with this one right here, if we were to do the multiplication, you know, you can carry through the thought here, the process here. And if it's an eigenvector, it should be some lambda times one minus negative two, it should be the original matrix. And I'm not claiming this is the eigenvector, it probably isn't. But can you check and identify whether you have an eigenvector for a specific eigen matrix or for a specific matrix, excuse me, don't don't ever listen to me when I talk about eigen matrices, I never I never said such a thing, it's a lie. And if you need some more practice, this one, you can go to section 6.1 of the textbook eigenvalues and eigenvectors, where we had some similar questions, examples and exercises, checking this very principle right here. Question number three is going to do with determinant calculations using properties of the determinant. So if you're given some information about the matrices, say A and B, in this case, we're given that they're both four by four matrices, we know some information about the determinants of A and B, A here is given as it's terminus two, B is determinant is three. And so then you're going to ask to calculate the determinant using properties of the determinant. So some properties that are going to be extremely relevant for us will be things like the following, the determinant of a product is equal to the product of determinants. Right, so the determinant factors in a manner similar to this. Other things we should know, like the determinant, if you take the transpose is unaffected by transposition, right, you get things. So the determinant of B is the same thing as B transpose. If you take the determinant of some scalar multiple of a, this is always equal to C to the n determinant of a, we're in this context, we're assuming a is some in my matrix, a common mistake that sometimes shows up here is that we sometimes think that the determinant is linear. And so we might erroneously think that this is C times the determinant of a, which is not the case whatsoever. The determinant is a multilinear function. And so you have to pull out factors from each row individually, not from the whole matrix. It's multilinear. This affects how one might try to take the determinant of a sum of two things. And so this is just these are just some examples of properties that determinants we've seen. Can we use those to calculate the determinant of these matrices here? The specific matrices will not be given just certain information about the matrices enough to determine the to calculate determinant, but you're gonna have these properties of these things. So if you need to learn more about properties of determinants, please turn to section 5.2 in the textbook, where there are some examples and exercises of those things as well. I always want to remind you that in the textbook, my students were only required to complete the spade, the spade questions, those that are marked with a spade, but there are plenty of extra exercises contributed by myself or by past students and such. Those exercises make good examples, good practice as you're studying for this exam. So take a look at those ones as well. Question number four, you're going to be asked to compute the orthogonal projection of specific vectors. And so if you need some review on that one, this is an older topic for this exam here. You can go back to section 4.5 for the specific formulas about orthogonal projection. But remember, as we're trying to do these orthogonal projections, this is this was using those linear combinations with the foyer coefficients and such, right? So if you have a if you have an orthogonal basis for your vector space w, the ideas that the projection onto w of the vector y, this will look like the sum where I ranges from one to say r, where you're going to take the vectors that are in the basis will take like u i dot y.  And so this fraction right here with the inner product on top and bottom. This right here is our foyer coefficients. They're the coefficients that show up inside of this orthogonal projection. And so then that's the scalar that's involved in these linear combinations, then you have to scale it by the vector u i here, which what is u i, our vector space w is going to be the span of these things right here, u one, u two, up to u r, things like that. And for this to be effective, this is going to have to be an orthogonal basis.  You will notice that in this example, there is only one vector that spans all of w. Well, as long as that's not the zero vector, which it ain't, you can see that right here, it's not the zero vector. Since it's not the zero vector, a single vector by itself does form an orthogonal basis. And so we can calculate the orthogonal projection. You want to know this formula, put it on your note card as you're preparing for this exam. And that's might be true for some of these other properties and formulas we saw in previous exercises, like those properties of determinants, you might want to put on your formulas as well, your formula shoot as well. And so then our last, well, I guess actually there's one more question to take that back. Question five on this first page right here is going to ask you questions about properties of, of eigenvalues with eigenvectors and such particular this question is coming from section 6.4 about symmetric matrices and Hermitian matrices and such. So for example, if we have a symmetric matrix, we are guaranteed that the eigenvalues are going to be real. That's also true for Hermitian matrices, we can guarantee the eigenvalues are going to be real. Another thing to look out for is triangular matrices. Remember, if you have a triangular matrix, which includes diagonal matrices, the eigenvalues are just the diagonal entries. So we can see that the eigenvalues of R are immediately one in three. Like so. So that's something we can pick up really quickly. Finding symmetric or Hermitian matrices should also be a very quick observation. So question number five is mostly going to be able to have you detect real eigenvalues really quickly on the other, you know, some might be a little more suspicious, but you should be able to find them really quickly. So this is not meant to be a super extensive type problem here. Like I said, if you look at the spectral theorem of symmetric or Hermitian matrices in section 6.4, you should be pretty okay for question number five here. Alright, and so now for the real, the for Rizzle last question on page number one, number six, you'll be asked to compute a determinant, it will probably be nothing too complicated, you see a two by two right here, I maybe ask you a three by three, which is not meant to be a super difficult calculation. If it's larger than that, like if I give you a four before, it's probably because it contains a large number of zeros, and you can cofactor expand across one of them, one of the rows that has a lot of zeros, so the calculations should be pretty quick. This is meant to be based upon things we talked about in section 5.1. Although we can use roll operations to help us calculate determinants in a more efficient manner, that's that's generally more useful for larger matrices. And you'll see a question like that later on in this exam. So this one should be just a quick one, like a two by two or three by three. And so you can use like these diagonal type of things to help us out here. Same thing for your three by three, right? So you have like ABCDEF, HGI, I hope I didn't skip a letter. Oh, I guess G generally comes before H in the alphabet. Whoops a daisy. But anyways, we've seen this thing before where you have this three by three matrix, you can always copy the first two columns, something like this, and then you look at the diagonals, you take this diagonal times plus that diagonal plus that diagonal, you'll add those ones together, then you subtract this diagonal this one, and I forgot one didn't I that one right there, there should be three going one way three going the other way. So the ones that go from the top left to the bottom right, those are going to be positive you add together. And those that go from the top right to the bottom left, or in this in these videos, they're blue right now, those are the products you're going to subtract. So again, these easy, easy, easy determining calculations two by two and three by three. Alright, and so the final question in the multiple choice here, it's going to be a question about cross products, can you do a calculation involving cross products whatsoever. So for example, this one right here, can you calculate that that cross product to you times three you, if the only thing we know about you is its length. So you should know the formula of cross products, you should also know properties of cross products, as they're listed in the textbook, in particular, you can want to go to section 5.4, if you want some more practice on cross products of any kind. And that brings us to the end of the multiple choice section. In the short response section, there's going to be three of these questions, there were six points each. Remember for the short response, the short response questions, please, please, please put your answer here on the line, right? This is where it goes. That's what it for. If you're a good student, and you want to be a successful here, put your answer here on the blank, right? That's what it's for. There is space provided. This is intention if you this isn't the point point of the space here is for scratch work. Unlike the multiple choice, there might be a little bit more to the calculation right here. You are welcome to put in the test, your scratch work in this area. You do not need to show work to get full credit. But if you do show your work and the correct and the response you put here on the line is incorrect, then you can receive partial credit with risk because of the scratch work that's provided here. So do do make sure you add some of that. Question number eight, given some vectors can we determine the angle between the vectors right? This comes from the dot product. We can take the dot product to help us find the angles between these things. So like there's the law of cosines that we've seen for vectors, which tells us that if we take the dot product of two vectors, u and v, this is equal to the length of u times the length of v times cosine of the angle between the two vectors. And so using this formula right here, we can we can solve for theta and do put this in degrees, the instruction to ask you for degrees. Unfortunately, if you don't have a calculator, I wonder what are you thinking? You can use a graphing calculator on this exam. But if you don't have a calculator, you can write it as like our cosine of whatever it is. Because at that point, I don't really care. But just for ease of grading, I'd prefer you to put in your calculator and give me the degree estimate and round it to like the nearest tenth of degree. Question number nine, you will be asked to compute the matrix representation of a linear transformation, giving two different bases, C and B right here. This is the type of thing we did in the very last section, 6.5, about linear transformations and similarity and things like that. I should have mentioned that the previous question with the law of signs, the last sign showed up in section 4.4 in the textbook. So you can go look there for some extra practice. But returning to question number nine, can you find the matrix representation given to bases and the direction does matter? This is going from B into C. And so remember exactly how do we how do we compute something like that? Well, I mean, if you want to, you can begin calculating the standard matrix. So if like A is the standard matrix of your transformation, so that would be using the standard basis of E's and such. We want to we have to compute T here with respect to C and B. We have to compute the matrix which has the property. I'm getting a little tongue tied here. Sorry, we're trying to get the well, I mean, I don't have it in front of me right now, but we do have the formulas in section 6.4 to talk about this change of basis formula. Basically, what we want to do is you're going to augment the matrix on the left. Sorry, the basis you're turning into on the left and on the right hand side, we're trying to look at what does A times this basis do right here. And so what I mean by this notation is kind of funky notation here because A is a matrix and B is a set of vectors. But what I mean is I want you to write each and every the image of each and every matrix going on right here. So you're going to get A times B one, you're going to get A times B two all the way down to A times B whatever in. So do all of those. That's if you have your standard matrix right here. If not, you can also just look at T of B one, T of B two, etc. Whatever. That's what we mean by here. This A of this A times B right here. Take the basis on the right act on it by the linear transformation which either use the make use the map T directly or you can use the mage the standard matrix that comes from it, whichever you prefer. And so taking that right here, C, which is the new basis, the basis on the left, A of B, A times B right there, you're going to reduce this thing. You're going to get the identity matrix here and then augmented over here is oh boy, there's not enough space. Slide it over a little bit. The identity and then on the right hand side, this will be your standard matrix here T with respect to B and C coordinates. Now I should mention that these matrix might not necessarily these matrices might not necessarily be square matrices. Like example, this one right here won't be a square matrix. So you might get rows of zeros popping up here. You can ignore those. Don't worry about that. Not a big deal. Like I said, you can see some more examples of this in 6.5. So do check that out if you have any questions. And that takes care of the short response section. That that's all there is that I was going to say about that. So the last six questions show up in the free response. Remember the free response. You do need to show your work in the space provided here work without it. I can't give you any credit. Even if you have the correct answer, I do need to see the work in order to give any credit. That's any credit for these questions here show all the work that's necessary. Question number 11, it's worth eight points. You'll be asked to compute a determinant. And this will be a larger determinant. We're talking like four by four, five by five, six by six, something like that. And this is in line with the type of questions we did in section 5.2. That is, you're gonna want to row reduce this thing, row reduce this to the RREF or some echelon form of this matrix. Because if you reduce a square matrix and echelon form, it'll be upper triangular. And therefore you can take the product with the diagonals. But remember, as you row reduce it, there's a cost that has to pay. If you do replacements, right, you add a multiple of one row to another. When you do replacements, these are going to be free. There's nothing that happened. Nothing you have to worry about. So replacements are free. If you ever do any interchanges, do be aware that it affects the determinant by a negative sign. So you'll change the sign of the determinant. And if there's any scaling, scaling, well, there's a way of handling. But personally, what I think of is like, when it comes to scaling, you want to factor, factor out the scalar from the row. And if you do that, you'll handle yourself just fine. And this is using the multilinearity of the determinant. And I do need to show all steps right here. As many of you will probably have a graphing calculator and calculating determinants of matrices as a standard graphing calculator function. I have to see the steps. Otherwise, I can't give you any credit, even if you write down the correct determinant. I have to make sure that you know how the calculation works and not your calculator. But by all means check your answer with the calculator. I'm perfectly happy with that. Question number 12, it'll be worth seven points. And so this right here, you'll be given a matrix, you'll be given an eigenvalue of said matrix. So you don't have to determine it. This is this is the real McCoy. This is a genuine eigenvalue for the matrix. And you'll be asked to compute a basis for the eigen space associated to this matrix. And this is something we did in section 6.1 of the textbook. What you have to do is you have to find a basis, find a basis for the null space of a minus lambda I, where in this case lambda is going to be the eigenvalue they give you. So lambda here is a two. And then a of course is the matrix they give you plug that in right there. So you take a minus lambda I you calculate the null space of that.  If you don't want to show the steps of row reduction, if you use a calculator to calculate the REF, I'm fine with that. I don't need to see the row reduction there. But once you get to the once you get to the REF, you need to pull out your basis. And as the eigen space, the eigen space right here, this is this is the lambda eigen space as the as the eigen space is a null space finding a basis for the eigen space, you have to use your calculation base, the basis of a null space, excuse me. And so while that is reviewed, we did talk about that in section 4.6 when we talked about all four fundamental spaces of a matrix. Originally, we had talked about it back in Oh, boy, when was it section 2.6 section 2.5 and 2.6 talk about this basis of finding the basis of a null space. We've done it many times, you might not need a review, but in case you do, those are places you can go in the textbook to find those things. Question number 13, you'll be given a matrix A, and you'll be given some eigen some information about eigen pairs. So that as you're given an eigen value like your negative one and a corresponding eigen vector, you'll be given another eigen value and it's corresponding eigen vector. And this can repeat depending on how big the matrix is. And so then given this information on eigen values and eigen vectors, you're going to ask to find the diagonal the diagonalization of the matrix A, which in particular means we have to find matrices P, D and P inverse so that A equals P times D times D inverse. And so when you're doing this, remember that the matrix P you're going to stick in the eigen vectors of the matrix, which as these are given to you, that shouldn't be too hard to find them, right? The eigen vectors are exactly things like this that were given to you. That's what we want to put in there. The next matrix is D. This will be a diagonal matrix for which your eigen values are given along the diagonals and then everything else is 0, right? These diagonal entries are going to be the eigen values. So please recognize that you have those eigen values given to you, right? So we're talking like here this is negative on this three.  And then the last matrix P inverse right here well to find P inverse since we have P, my recommendation is to just apply the inversion algorithm that we've seen before. How do you find the inverse of a matrix? The inverse of a matrix we had talked about this in section 3.4 for the inverse of a matrix. You want to compute that. Now again your standard graphing functions have a function for computing inverses of matrices and I do want to see a little bit of that. What I would want to see here is you're going to take P augment the identity row reduce that into the identity augment P inverse although I don't need to see the row operations you can omit those here but I do want to see I want to kind of see that you understand this principle of it and then that will give us P inverse right here. So I do so given information about the eigenvalues, eigenvectors of a matrix can you find its diagonalization this is the type of stuff we were talking about in section 6.3 entitled diagonalization you will not be required to do the whole enchilada but you do have to wrap the tortilla so to speak the last part of it you have to know how to do that's question number 13 number 14 should be down here can we solve the least squares problem so you'll be given a linear system of equations it's probably inconsistent most definitely I'm going to make it inconsistent and I want you to solve the least squares problem find this least squares solutions which there could be multiple solutions so you'll be given this system AX equals B remember to solve the least squares problem you switch over to the normal equations see times left side the left and right by A transpose and then you're going to solve this linear system the solutions to the normal equations give us the least squares solutions this X hat and then you report X hat there if there's a unique solution tell me what it is if there's multiple solutions write the general solution the general least squares solution and so this is what we talked about in section 4.8 the last page we have question number 15 it's worth 7 points given a matrix you'll be asked to compute the eigenvalues of said matrix my recommendation is you want to compute first the characteristic polynomial the characteristic polynomial of the matrix and so remember the characteristic polynomial is you take the determinant of A minus lambda I treating lambda as a variable right and you know sometimes if if you wanted to you could take the determinant of lambda I minus A that's I prefer this 2 by 2 matrix it makes no difference but there they'll give you the same things that's that's what we have to do calculate this this will then give us some polynomial right f of lambda it'll equals you know for this is a quadratic one so it looks like some a lambda squared plus B lambda plus C something like that set it equal to zero and you're going to solve for lambda lambda equals yada yada and then those are the eigenvalues you're looking for so use the characteristic polynomial this is in line with exercises we did in section 6.2 alright question number 16 here number 16 this this question right here is a question about the Graham-Smith process Graham-Smith which the Graham-Smith process remember you you have to find an orthogonal basis for the for the sub basis provided right here so you have vectors x1,x2,x3 you want to orthogonalize that basis this is the type of stuff we did in section 4.7 so look there for some more practice remember there is an algorithm for this it's recursive like your first vector you don't do anything to it you just leave it alone for the second vector v2 you'll take the second vector and subtract from it it's orthogonal projection which I'll just write it out like here we just take v1.x2 over v1.v1 times that by v1 like so then v3 would equal x3 minus v1.x3 over v1.v1 times v1 then also subtract from that v2.x3 over v2.v2 and don't forget to put in the v2 over there and then you continue doing this if you have more and more and more for real vectors it's not so bad but for complex do remember the order of your Hermitian product does matter so make sure that the x is showing up on the right and the v is showing up on the left to get that correctly and this will help you find an orthogonal basis if you have to find an orthonormal basis do be aware that you have to normalize the vectors the Gram-Smith process does not necessarily give us unit vectors but we can always normalize when this thing is over with so do make sure you look for that before before you make sure you're done this one only is this one only asks for orthogonal basis so it's not a big deal but be aware that it might be orthonormal it might be orthogonal depends on the instruction right there and so that gets us to the end of the exam oddly enough and I apologize I think I forgot a question so we're going to go back up in the short response my spider-sense has been tingling that I only did two questions and I think it's probably because I forgot to slide to the bottom of the page with the short response so if you've been paying attention number 10 was missed I apologize for that question number 10 in the short response is going to be a question about the left null space of the matrix so a matrix A will be given to you and you'll also be given information about if you have the matrix A augment the appropriately sized identity matrix this will reduce into this thing right here you are expected to find a basis for the left null space and if you take A augment the identity and reduce it to some REF like we see right here this is in fact row reduced you have the pivots here in the first and second column this is a this is an REF of A augment I and you can honestly get away with any echelon form but to find a basis of the left null space all you have to do is grab these vectors on the right hand side that coincide with rows of zeros and so then your basis will just be those two vectors right there that's fine or column vectors whatever whichever you prefer in that case so whoops a days we almost missed that one but that shouldn't get us through all 16 questions on this exam and so this does cover the the main important topics from chapter 4, chapter 5 and chapter 6 about orthogonality determinants and eigenvalues and this is going to be our last exam for the semester as always if you have any questions feel free to reach out to me you can either of course just leave comments in the video below I'll be glad to answer them there or you can come to office hours send me an email and I'll be glad to help you as you're studying through this test remember the solution to this test can be found at the end of the exam and the test that you will see for exam 3 will be a very similar structured test to what you see in front of you right now alright everyone best of luck in your studying and I will see you later have a great day everyone bye


Every person in the world would like to have more happiness in life. Our sages teach us, our rabbis teach, that it's a great mitzvah. It's an important thing for every person to be happy as much as possible. And in our society, throughout Canada, in the United States, in Europe, there have been many books written just over the past few years, many books written about the topic of happiness. And one of the things that has been discovered in almost all of the studies, people, doctors and scholars and professors that study this topic of happiness, one of the things that almost everyone has discovered is that having a spiritual, religious life is a very important ingredient in being able to have a happy life. Having lots of money is not something that necessarily leads people to being happy. And the truth is that often, when people have a lot of money, they're not very happy. There's no direct connection. But there is a direct connection between being a spiritual person and being a happy person. So what I'd like to discuss this morning is how can we all work towards having more happiness in our lives? The truth is that there is a tremendous amount in life, in each of our lives, to be happy about and to be thankful for. If we really think about it, we each have a tremendous amount of blessing in our lives. And one of the greatest problems that gets in our way is that we are often our own worst enemies and we sabotage, we destroy the happiness that we could have by failing to appreciate those blessings. Now to be truly happy in life, we have to be true to ourselves. If we're going to be really happy, we have to be real. We have to be real. We have to be true to our real nature. We have to be true to our real nature. What does this mean? The truth is that every human being is physical. When God created human beings, he created us from the earth, from the dust of the ground. So a part of us is very physical. We like to eat. We like to sleep. We like to do the things that animals do, horses and cats and dogs. We're very physical like everything else on the planet. However, there is another side to us which is not just physical, it is spiritual. Because when God created the first people, the first human being Adam, God breathed the breath of life into that first human being, which is our soul, our nishama. So we have these two parts to who we are, our physical side, our spiritual side. And the truth is that if we only focus on our bodies and ignore our souls, we're not really living as human beings and we will never be truly happy. You can't be truly happy if you're not being who you are. And so the important part to remember is we are not just physical, we are physical and spiritual. Now Judaism teaches that we are in this world to grow. That we were put into this world in order to grow. Which means to work on ourselves, to work on our medot, our character traits, which means what? It means when we go through life, trying to be people who are more sensitive to others, trying to be people who are more generous, trying to be people who are more calm, trying to be people who overcome our tendencies to be jealous or to be impatient or to get angry. These are all parts of our personality and we were put into this world to work on them, to develop them, to grow, becoming better people and working on our characters can bring us tremendous pleasure. It's like building anything. If you see a child that takes some blocks and they build a fort or a castle, they enjoy building. We as human beings, we are not happy when we're doing nothing in life if we're wasting our time. But if we build something, if we create something, we feel great. An artist who paints a beautiful picture, steps back and looks at it and feels wonderful. Look what I accomplished. So each one of us is an artist. Each one of us is an architect. Each one of us is a builder and we're building something that can be beautiful and magnificent and wonderful. What are we building? We're building a human being. When God created us, God said, let us make man. God said, let us make man. Who was God speaking to? God was speaking to you and me. God was speaking to every human being and God was saying, I cannot make you. God made every other thing by just saying, let there be trees, let there be animals, let there be fish. God can make everything. But God is saying, I cannot make a human being.  I can give you the raw ingredients, but only you can make you. So that's why God says, let us make man. We become partners with God and we have the ability to create a masterpiece. And if we do that, if we are able to step back and look at our lives and say, you know what, last year I was someone that got angry very easily. And this year I worked on myself and now I'm much more calm. I feel wonderful. I feel very, very happy. So this is one of the ways that we're able to become happy by growing ourselves and becoming bigger people. And that's why in Hebrew the word for growth is related to the word for happiness. The word for growth is sameach and the word for happiness is sameach. So the Hebrew words for growth and happiness are really pretty much the same. When we grow, we become happy. Another teaching of the Torah about how we can achieve the greatest pleasure possible is to connect ourselves to the greatest thing in existence, to connect ourselves to the greatest reality in existence, to connect ourselves to the source of everything that exists. And that is Hashem himself, God himself. If we get close to Hashem, if we get close to God, there is no greater joy that's possible. When we think about it, our world is filled with beauty. We go out in a beautiful sunset. We can see the beautiful sunrise. We can see the stars. We can see forests and lakes and oceans. There's so much beauty in this world. And this world contains so many treasures and so much pleasure. You can find a delicious fruit and eat it and say, oh, this is wonderful. And the truth is that we have the ability to have a relationship with the creator of all the beauty and pleasure in this world. When I was a teenager, so like any other teenager, I loved to nash on cookies and cake. So I got a job when I was in high school at a factory that produced cookies and cake. It was amazing. This is not just buying some cookies in a supermarket where they were sitting on the shelf for half a year. I was able to eat the cookies right off the assembly line. They were so fresh and they were hot and they were delicious. The closer you get to the source of things, the closer you get to where it comes from, the better it is, the more delicious it is, the more pleasure there is. The same thing that we live in this world and the closer we get to the source of everything in the world, we can have the greatest pleasure. People today will line up. They'll stand in line for hours to get the autograph of some famous baseball player. But God is close to every single person that calls to him. The Torah says that God is near to everyone who calls to him. Try to get a personal appointment with the Prime Minister of Canada. Try and arrange a meeting with the President of the United States. Try to get to see them. It will be almost impossible. Almost impossible. Or it may take you a year or two years. And maybe you'll get to spend three minutes with the Prime Minister. Maybe if you're lucky. But we can speak directly to God whenever we want. And God listens to us. And God actually knows each one of us and knows our name and cares about us and loves us. If you can appreciate your closeness to Hashem, your closeness to God, you will have everything. Of course, life is often very difficult. Life is often not so easy. And one of the ways to maintain a positive attitude is to have trust in God.  And he explained how he was able to survive this terrible experience of being in prison. He said that he once saw the most amazing thing. He saw a man that stretched a steel wire between two tall buildings. And he climbed up and he walked across this wire. Very, very dangerous, very amazing. And many people were looking up and they were watching. This is incredible. Look at what this man is doing. The next day he climbed up to this wire again. But this time, when there was a big crowd down below looking at him, he said, before I start to walk across the wire today, I'm going to tell you I'm doing something even more difficult. I'm going to take a wagon and I'm going to put the wagon behind me. And I'm going to drag the wagon behind me on the wire. Everyone is amazed. But then he says, is anyone willing to get into the wagon? So a young boy steps up and says, I will get into the wagon. And the man walks across this steel wire between these two buildings very high up. And he gets all the way across. And the people down below, they can't believe what happened. And when the man and the young boy come down, everyone crowded around the young boy and said, why were you willing to get into that wagon? Who are you? So the young boy said, that man is my father. And I know that he would never put me in a situation where I would be in danger. I trusted him. But then everyone came over to this man and said, how were you able to walk across this wire, this dangerous wire, with your son dragging behind you in the wagon? How did you do that? So he said, I'm going to tell you the secret of how I was able to do it. When I walk across that wire, he said, nothing in the world exists for me. For me, nothing exists except the next step I'm going to take. There isn't even a wire. He said, if I was to look down, or if I was to look ahead to see how far it is to get to the building, he says, I would lose my balance and I would fall. The only way I was able to make it is I only thought about taking the next step. And the rabbi explained, that's lesson that he learned from this man is what allowed him to exist and not just exist, but to survive in a healthy way all of those years in that prison, that terrible, terrible prison. He said, I didn't worry about the past. I didn't worry about the future. I didn't worry about how long I'm going to have to be in this prison. He said, all I thought about was, what does God want from me right at this moment? I just thought about this moment and about being with God and the fact that God was with me and I was never overwhelmed. I never felt terrible. I never felt sad. Another very, very powerful tool for happiness is learning to be thankful for those things in life that we have and not to focus on those things that we don't have. Our rabbis teach that the happy person, the person that's really wealthy, is the person who is happy with what they have. If you are happy with whatever you have, you are a happy person. It doesn't make a difference how much you have. And that's why, what is the holiday in the Torah? What is our holiday? That's the happiest holiday? The time of our happiness? It's the holiday of Sukkot. What do we do? We don't stay in a beautiful house with beautiful things and fancy things and expensive things. We build a little hut. A little hut we build in the back of our house. Very, very simple. And we sit there for a week. And we have the simplest things in life. But we realize that we have everything. We have everything. Even though it's not much. So when we get up in the morning, we say, I thank you God for the fact that I'm alive. The first thing we do is we say thank you God. I'm alive. I appreciate this. If we can't appreciate being alive, we'll never be happy. God created human beings similar to God. God is intelligent. Human beings have intelligence. God has free will. Human beings have free will. As I mentioned before, God created the world. But we are creators. We create ourselves. We can be very similar to God. God asks us that just as he is holy, we should be holy. We have that ability. We have the ability to be like God. One of the ways we can do this is God is a giver. God is a giver. God is not a taker. When we think about God, God is always giving. People will tell you, people that are happy people will tell you that one of the things that brings them the most pleasure and most happiness in the world is when they are giving to others. When they are helping others, it makes them incredibly happy and fulfilled. It doesn't have to be a lot. You are just smiling at someone. When I go to the supermarket here and you see the people that are behind the counter and you pay them for your food or whatever you are buying, they stand there for hours each day and no one is nice to them. They always seem to be so unhappy.  They look happy. You just made someone happy. It didn't cost you anything. It was very easy. So just smiling at people and being friendly and saying hello and complimenting people. Thank you for the job that you have done. You did a great job. Helping people, making other people feel good, makes us feel good. When we stop being concerned only about ourselves, when we stop being selfish and we try to give to others, it will make our lives happy and joyous. There was a great rabbi who received a letter from someone who was complaining about how they were not happy in their life. And the person was saying that I am not happy and I don't feel that my wife really respects me and I don't think that my children really care about me that much and I don't feel that my boss at work is really respecting me and this person went on and on and on complaining to this great rabbi and he said, why can't I have a life that's happier? And all the rabbi did was take out a red pen and he circled each time on this letter the person wrote the word I. I. I. I. I. I. The person, the rabbi was trying to say only thought about himself. It was only me, me, me, me and such a person will never be happy. We can become bigger people by giving to others and becoming bigger people will certainly make us much happier people. It feels great. It feels good to give. I want to thank you very much and I hope everyone has a much, much happier life.


Thank you very much and yes, thank you for having me. So my name is Björn Wenzel. I am part of DB Schenker. And yeah, I will talk a bit about how we migrated clouds with zero downtime. And at the end, we as Schenker do not only deploy containers, we also ship them. So it started at the end with an existing AWS Cloud infrastructure. We had in place managed by a subsidiary of Deutsche Bahn. We had not the ability to transfer from an AWS Cloud to another AWS Cloud or simply an account transfer to another root account or something like this. We had to change in the IP ranges. We had a highly restricted environment from a port point of view. Every port has to be requested, dedicated through a firewall. So all these shitty things you would like to deal with, especially when you're in the middle of a migration and you have a time pressure. And for sure, lots of day-run operations, which means like, yeah, we tested it. It worked. And yeah, maybe let's go now and get more standardized. So our target setup was in a cloud landing zone that was developed together with a partner company and set up from scratch. For sure, highly automated infrastructure, much larger IP ranges, much more flexibility and dedicated accounts per team. Because before they were shared. And for sure, the general rule new is always better. So general considerations we had in mind was for sure we don't trust our network. We have a general encrypt everything policy, our encryption everywhere policy. And we would like to go also with zero downtime migrations, even during business hours or customer times. So our source environment, we already played a bit like our own service mesh approach, or at least the TLS part of it, and some kind of observability we got already out of it.  This onboard proxy was configured to take from a secret the SSL certificate. And if this certificate changes, because we had an operator in place called vault CRD, that fetched from hashicob vault a new certificate when the certificate was near the expiry date. So we fetched a new certificate, placed it in a secret, and when the onboard detected that there's a new secret in place, it simply performed the hot reload, which was the cool thing. And at the end, another application that was running in the same cluster could simply connect to our application via this onboard proxy. Simply had to trust the certificate authority we had in our hashicob vault. So this was our setup before. So we looked a bit around and said, yeah, what are the potential alternatives? Or what are the ways that could support us with this cloud migration? So we came up with Istio on the one side. But to be honest, for me, it was very highly complex. It is continuously changing, in my opinion. At least at this point in time, there was a missing clear governance, in my opinion. Consul as an alternative was very alpha at this point. Submariner, we have not really considered further. But then we came to Linkadee, and at the end, the approach of the multi-cluster they have is, in my opinion, quite simple. And it's based on an Nginx container at the end, which acts as a gateway and the usual MTLS related topics. So maybe as a short overview, the Linkadee multi-cluster approach is simply you annotate a service in the target cluster, and then there is the Linkadee magic happens with the Linkadee service mirror. This watches on the API, creates a, I call it, federated service, so a copy of the service, and points this copy of the service to the Linkadee gateway. And when I now talk via a Linkadee-enabled application to the federated service, the traffic goes through the Linkadee gateway to my target application. And this is what we at the end utilize simply. So we replace at the end our application simply with the Linkadee proxy and enabled on the other side on the target account that this service should get mirrored. And the interesting fact out of this is that for my application, that is running on the left side, it was completely transparent that the traffic now goes to the other cluster. It has not to care about this. It was simply the request was still arriving with the old approach of the Onward proxy and then the Linkadee proxy, which was configured to only take up outbound traffic and not inbound traffic, took it up and forwarded the request to the new target application. Yeah, we only had to open two ports. As mentioned, we have the topic of firewalls, how to handle firewalls, et cetera. And final words, because it's very fast, it's very short time. So what is definitely needed is a very good monitoring of this, about the linking. If you run it for a longer period of time, how to handle it is definitely an interesting question. In general, for sure migration not only contains HTTP traffic migration, but it also contains topics like database migrations or Kafka data migration, et cetera. So these are things that needs to be considered. And at the end what started with a migration topic for us, we needed a tool for migration now ends up with our service mess of choice. And what we found out is also very interesting. We simply send a merge request and it was very fastly accepted. So thank you very much. Thank you. Given that we only have a few minutes to turn around for our next lightning talk.


Everybody forced here hoping 2023 gold standard full case number four Random team style here are the results on your screen. Let's get started everybody Link for the results in chatting find them anytime late in sports card calm here. We go here. We go here. We go Half foot away from the goal next year for gold Sam dressing up as a prospector Strength of your team is not the running backs They run Josh Allen all the time they're They're gonna do it here and score probably but still so 99 we have Daniel Jones 75 a rookie Dalton Kincaid for Buffalo Aiden Hutchinson jersey card out of 199 for the Lions. Let me get some fixes. Yeah sleeve all the thick stuff Hudson to 199 Dual patch rookie newly minted is Bijan Robinson a Jamir Gibbs to 49 That'll be a random at the end between the Falcons and the Lions. Good luck on that one Gold gear jersey to 299 Kenneth Walker the third Seahawks Garrison Hurst on the Midas touch auto number 9 of 49 for Arizona Garrison Hurst and a Jaguars rookie autograph tank big speed 12 of 49. Oh, what do we got here? We got an AJ Brown the 99 for the Eagles. What is this? What is this? It is opulence one-of-one rookie Packers Jaden Reed Wow, Kyle and with the Packers I've never seen an opulence one-of-one rookie whoo looks so sick with that platinum foil, too I miss opulence basketball a lot. Oh my god That is sick That is sick George Pickens mother load Jersey card to 299 Steelers Tankdale newly minted rookie Jersey to 399 Texans Purdy gold gear three of 10 49ers You have to do some work man Lost the best receiver and then McCaffrey might be injured too Jonathan Mingo triple Jersey rookie auto Panthers out of 149 and to cap off a fantastic box is an RPA of Bijan Robinson to 49 Falcons 26 of 49 of the Bijan Let's get that one magged Nobody nobody cares about the one-on-one. I don't care dude Who's watching the bill's attempt to score a single point to 99 Breeze Hall Jets? Rookie to 14 as Marta Mappu for the Patriots. I mean, we'll see Eddie We'll see LaVisca Chanel to 299 Jersey card for Carolina Who scored was it was it Josh? Oh someone's hurt. Who's it throwing throwing to? Shakir Hardy Hardy Ty J Spears rookie newly minted Jersey to 399 for the Titans Dak Prescott mother load patch card to 49 for the Cowboys Steelers great hit here one of five gold fingers autograph Kenny Pickett Steelers great second-year picket auto again one of five on that thing. We'll get the picket in a mag and A Jalen Carter Eagles dual patch rookie out of 10 of 49. It's been a good case so far only three boxes in Two to go on flawless everybody just to Still 17 on the mixer About halfway to gold packs for spot plus Emmett Smith signed helmets Random to somebody in that break to every spot. I'll be ran into every spot One lucky person in the break. We'll get the Emmett Smith authentic signed helmets Before you can open a box to 75 is sake one for the Giants Rookie to 99 is Brian Branch Lyons Chiefs gold mine Jersey to 299 is going to be sky more book and serial numbering there Dorian Thompson Robinson rookie Jersey to 399 Browns White gold Bengals patch card Jamar Chase 3 of 24. Here we go CJ 2k all-commissed autograph to 25 cool hit for the Titans And a Jake Hainer rookie Jersey autograph to 1 99 Saints or not a card stands for these hits All right, rookie to 49 Stetson Ben at the fourth for the Rams to 99 Alec Pierce Colts Sky more gold mine Jersey number to 299 for the Chiefs again Justin Jefferson gold mine Jersey to 299 Vikings Colby Wooden rookie autograph out of 24 Packers Good as gold Jersey autumn Alcum Rodriguez to 1 99 go into Detroit and Dorian Thompson Robinson rookie dual patch autograph out of 49 for the Browns Solid QB hit there for the Browns. Let's keep it going gold standard. Good case. We got ripping here Box six Number to 75 is a con with dolphins and a 75 Trevor Lawrence Jags Jersey card at 299 Leviska Chanel for the Panthers Dual patch to 24 rookies will Anderson jr. And Tyree Wilson That'll be a random at the end Texans and the Raiders White gold Justin Herbert Jersey card at 299 Chargers Got Trenton Simpson rookie autograph out of 24 for the Ravens and nice a Josh down to 24 RPA Colts He's been very good 19 at 24 Colts RPA Josh Downs Like a line right there. It's weird Guess that that's like a scuff.   Let's go a fantastic promo tonight white gold Jersey to 299 Isaiah Pacheco for the chiefs Tank Bigsby rookie Jersey to 399 for the Jaguars our pretty goldmine Jersey to 299 Niners Luke schoonmaker Ricky dual patch out of 49 Cowboys And Lions agent mere Gibbs Ricky dual patch autograph to 149. That's you here. Keep it rolling Spencer Eddie Sean to me. That's that's who's all here Crazy Ricky to 49 Tavius Robinson for the Ravens 24 James Connor Cardinals Alec Pierce gold rush Jersey to 299 for the Coles Chase Brown Ricky Jersey to 399 going to the Bengals got a Garrett Wilson gold gear Jersey to 299 jets Good as gold sky more Jersey out of the 199 chiefs Lot of sky more in the break And a zave flowers Ricky dual Jersey out of 149 Ravens had a really good really good game today All right, rookie to 49 as Makai Blackman for the Vikings and a Murray Cooper to 99 for the Browns Mark how's it going man? How's your trip? Cam makers gold gear Jersey to 299 for the Rams Chargers Echler Jersey to 299 Jackson Smith and jigba triple relic auto rookie to 149 Seahawks While we're refreshing. Hey dude, I hear that. I'll take that trip any day Gold strike autograph for the Chargers Rodney Harrison to 49 Chargers Rodney Harrison At least the baseball game is meaningful Anthony. You're back. Oh, thank God. Oh, so worried Doreen Thompson Robinson triple patch rookie out of the 49 Browns Other DTR Sean was like, where's Anthony man? When is he gonna come in chat and start saying all this cool stuff? Let's go Giants They're run of the ball. Well, that's for sure. So this drive All right, rookie to 75 J. You're brown Niners got Mack Jones to 99 for the Patriots Titans mother load to 24 five-way relic trail on Berks Josh Downs Ricky Jersey to 399 for the Colts here Justin Jefferson goldmine Jersey to 299 for Minnesota. There's rookie autograph Tyler Scott a number to 199 on the Scott And a Quentin Johnston dual Jersey rookie autograph to 149 Nice Quentin Last two boxes here So 99 is Jalen Hertz Eagles number to five rookie Jameer Gibbs Congrats to the Lions. That's sick Andre with a spot on Sunday fun day Paul with a spot on Sunday fun day bringing us down to what 15 now Get that one Mag for your Lions 15 to go on Sunday fun a pass the halfway point Again, everybody Emmett Smith helmet and two gold packs per spot Emmett Smith help We're in everybody to break two gold packs per spot though Craig Reynolds. All right DJ more on the patch card to 49 for Chicago. They're Penini They're national VIP gold packs by Penini So they have their chrome cards in there. They're kind of like prism pack They're kind of like prism almost different different design, but multi sport packs with numbered cards in them sometimes autos as well Henson Hooker a rookie patch card to 399 for the Lions was like multi sports I was like racing basketball football baseball. I think there's some soccer in there UFC wrestling One of our customers hit like a stone cold autumn number to five out of one of them There's a ton of other crazy stuff in there, too. That's St. Brown Jersey to 299 for the Lions Sun God Mackay Blackman on the rookie autograph out of 199 for Minnesota Yeah, we were why we were watching a Spencer. We saw that one got a rookie auto to 199 Josh Downs with the Colts That is true. There's true Spencer hates his first round pick last one Got a couple randoms to do All right, you got rookie to 24 Anthony Richardson congrats Colts It's a 99 Stefan Biggs for Buffalo DJ more again Jersey card to 299 bears Sup Rory Jamiro Gibbs rookie patch to 24 nice card Lions back Prescott Jersey card to 299 for the Cowboys Could have just kept you Andre Swift Spencer Gold strike Saints autograph Marquez Colts into 99 and Josh Downs again to 24 Colts Rookie jumbo patch autograph man Colts did really well Yeah, I agree James Wemby Like even just the glimpses of like what he could be and like he already is it seems like he's gonna be pretty good this year but man the potential that kid has is Literally something we've never seen before dude. He had a dunk the other night and like transition Like barely like he didn't have to jump like it was like running still like so far away and still don't yeah Like you some of the things that he's gonna do you cannot stop. Did you see the reaction of the defender? Walking away like like looking around like dude, what am I supposed to do there? Can't do anything. He's literally gonna do whatever he wants them There's gonna be times where he's like doing things like that and it's like there's no defense for it Except somebody that size but how many people that size can move like that. Nobody is actually scary Very cool to see that like very cool to see. All right guys want to go on flawless What's up Victor? Laying on landing on thick right there All right two dual relics here random to both the teams on them obviously first one is gonna be falcons lions Robinson and Gibbs the next one is Texans and Raiders Anderson and Wilson luck to the teams involved We'll go six times on our random randoms whoever's up top get some one Two dude, I'll trade you Mac for field straight up right now. So quick or are you have whatever dude? Yeah? No, I like fields three four five and six at least comparatively Falcons Falcons But I wouldn't wish anybody to play quarterback of the New England Patriots right now and Then six times on the other one one Two three four five and six the Texans Nobody should have to go through playing quarterback of the New England Patriots right now. That's just that's just mean It's like something you get sentenced to give their relics. Yeah Recap here we got relics of Dak from your Gibbs the 24 is a nice one DJ more almond raw Hinden Hooker DJ more Justin Jefferson trail on Berks Josh Downs Cam Acres Austin Eckler Garrett Wilson and Chase Brown Alec Pierce Tank Bigsby Brock Purdy Isaiah Pacheco Justin Herbert Levisca Chanel Justin Jefferson Skymore Skymore More Skymore Doreen Thompson Robinson Jamar Chase Taijai Spears Dak Prescott Levisca Chanel Brock Purdy to 10 George Pickens Tank Dell Aiden Hutchinson Kenneth Walker the third Autos we got Jeter Downs Marquez Colston Josh Downs I said Jeter Downs dude. I said Jeter Downs You've been saying Jeter Downs a lot over the summer Josh Downs. I know Josh Downs is Josh Downs a beast Josh Downs twice there though Mackay Blackman Tyler Scott a Quentin Johnston Dorian Thompson Robinson Jackson Smith and jigba Zay flowers Rodney Harrison for the Chargers Skymore Jamir Gibbs We have Luke schoonmaker Josh Downs again Trenton Simpson Colby Wooden Dorian Thompson Robinson Malcolm Rodriguez CJ 2k Jake Hainer Jalen Carter Jonathan Mingo Garrison Hearst Tank Bigsby You need one of one opulence of Jaden Reed for Green Bay. That's a sick one-of-one there RPA Bijan a 49 for the Falcons Number to five is Jamir Gibbs for Detroit and Goldfingers second-year auto Kenny Pickett one of five for Pittsburgh Thanks again everybody. We're going on now to flawless