 Welcome to our exam one review for math 2270 linear algebra at Southern Utah University. As usual, I'll be your professor for this review, Dr. Andrew Misseldine. What you see on the screen right now is a copy of the practice exam for exam one in linear algebra. This practice exam is going to highly resemble the exam that we will be taking in just a little bit. And so I want you to use this practice exam as a way of preparing to take the actual exam. Now be aware that this practice exam is not a collected assignment. It's not graded. There are solutions to this exam. There are there's paper solutions attached at the end of this document. Plus also you can find some solution videos here in this channel as well. Now when you first get the exam, what you're going to do is you're going to see this cover page that looks something like you you see in front of you, right? Make sure that you write your name, of course, on the exam so that you get credit for the work you've done. There will be a list of instructions on the front page. This does change from time to time. So I wouldn't take what you see in this video too seriously. Because again, from semester to semester, there are some needs to change some specific instructions or what such. So do refer to the information on the learning management system that is the practice the practice exam that's distributed by me on Canvas or wherever it's located. Do do follow those instructions. Now some things I can mention that are going to be true. So you're going to see this table at the very end as well. You don't have to worry about that. This is just a place where I'm going to fill out information for you as I grade this exam. Now some things to be aware of there will be 15 questions on this exam. The first category of questions is what we call the multiple choice section right here. On this exam, there'll be seven multiple choice questions, and they're worth five points each, given as a total of 35. With multiple choice questions, they are pass or fail. That is, you either select, you select the single correct answer, and if you do that, you would get the full points. If you do not select the correct answer, or if you select multiple answers, or your answer is not clearly indicated, then of course you won't get any credit for these multiple choice questions. You've got either the full five points per question or zero points. You are not required to show any work on a multiple choice question, although it might be a good idea to work out some things. The multiple choice questions for the most part are on the easier side, and probably don't require a lot of steps. They're meant to be simpler type questions. The second category of questions, and we'll see some examples of this in just a little bit. The second category of questions is what we're going to call short response questions. They're very much on par with the multiple choice questions, but unlike multiple choice, no potential answers will be provided to you. Instead, there'll be a blank that's provided and a prompt, some question that's given. You'll reply to the question by writing your answer on the line or in the blank that's provided. With the short response questions, to get full credit, all you have to do is write the correct answer on the blank that's provided. If you write the correct answer, you don't have to show any work. You can get the full points. On this exam, I believe there are two short response questions. They're also worth five points each. Now, if you're, and if you so, if you write the correct answer in the blank, you'll get the full five points on those questions. If, for some reason, you do not write the correct answer on the blank, you can receive partial credit for the short response section. That is because maybe your answer that you write is partially correct or there will be a space provided below each short response question where you might do some scratch work. You can receive partial credit for partially correct work provided in that area. Now, like I said, you do not have to show your work on that one, but you can cover your bases a little bit by making sure you show adequate work in case, again, there was a mistake that was made. Many of the short response questions won't really have a lot of work that has to be shown. Again, kind of like a multiple choice, you don't have to show your work, but you can get some partial credit from that. Now, the remaining questions on this exam, questions 10 through 15, so that's 16, that's six questions right there. These are what we call the free response questions. Now, unlike the multiple choice and the short response, you must, you must show your work in order to get full credit in the free response section. The free response section consists of questions which are much more longer and challenging, and therefore, in order to get full credit, you have to show your work because that's what I'm looking for. I mean, the final answer will be helpful. Yes, but I want to see the intermediate steps. That might mean something like showing your row reduction, like doing your elementary row operations and illustrating those things. It might mean giving an argument that involves sentences or just showing your work. Now, not all the free response questions are created equal. You'll notice that while many of them are worth 10 points, that's the maximum pointage for a question. Question number 10 is worth eight, and question number 12 is only worth five points. They're just, you know, they're not full. They're not the full 10 points because those ones are a little bit on the easier side. Now, you will notice that on question number 16 here, this is mostly just a placeholder. It's only worth two points. This is for submitting your note card at the end of this exam. You'll submit that with the exam. You are allowed during this exam to use a calculator, which includes a graphing calculator that's perfectly acceptable. I understand that graphing calculators typically have some type of matrix operations, for example, like row reduction. So the calculator could be very helpful in checking your work. And so I do allow graphing calculators. If you do not have a graphing calculator, you can use another type of calculator like a scientific calculator. You can use a four-function calculator to help you with arithmetic and things like that. That's the main use, but graphing calculator is appropriate. You cannot use your cell phone, even if it has a calculator app or any other type of device. It has to be, its sole purpose has to be a calculator, standard calculator like Casio, Texas Instruments, something like that. So you are allowed to use a calculator. You are allowed to scratch paper, of course. Be aware that the scratch paper will not be collected. Only the stuff you have written in this test pack is what I want to see. And then also that you're allowed notes, you'll turn in with the exam. And that's worth two points. If you didn't have any notes, then of course, you won't get those two points. So make sure you do that. So those are some of the policies I want to mention about this exam. And so with that, let's move on. Let's see. Oops, I know to the next page. So let's take a look at the questions that are available on this test. And like I said, there might be some specific policies about this exam that, again, they might change from semester to semester. So I don't want to delve too much into that in this video. You can find that information on Canvas. Please read that to make sure, like, is there a time limit? Where do I take the test? Again, that changes. That changes because of the needs of the semester. So look to Canvas for that information. So let's start off by looking at the multiple choice questions. The first multiple choice questions. And I should mention that what you see in this practice exam is not going to be verbatim what you see on your actual test. Now, I did design the practice test and the actual test to be very resemblance of each other. I want the practice test to be a good example of what you could see, but there will be some randomness to it. So don't expect exactly the same things. Just because something is included on the practice test doesn't mean it will necessarily be on your actual test. And just because something is omitted on this practice test does not mean that something might not be included on the actual test. In terms of coverage for this exam, this exam covers the first third of our course, which goes from section 1.1 all the way to section 2.5. So this includes all of chapter 1, which was our introduction to linear algebra. So we have our introduction to linear systems. We have our introduction to fields. So in addition to the real numbers, we should know how to do arithmetic with the complex numbers, working modulo of prime, like mod 7, mod 2, mod 5. We should know the axioms of the vector space. We should know how to do algebra with column vectors like linear combinations. This first question really is in line with that. We should know about linear transformations. How do you compute the kernel of linear transformation, the image of linear transformation? How do you improve it to linear transformation? We saw that in section 1.4. 1.5 and 1.6 were about augmented matrices, echelon forms, and row reduction of matrices. Then we started chapter 2. This test will not cover all of chapter 2, but it will cover the first five sections. We have section 2.1 about the vector equation, 2.2 about the matrix equation, 2.3 about linear independence. We should be able to determine whether a set of vectors is independent or dependent. In 2.4 and 2.5, we introduce affine geometry and subspaces. We should know all about those things for this exam right here. Like I mentioned, in question number 1, it gives you two vectors, vector u and vector v. These are vectors that apparently are in R2. One thing I want to mention is that when you see vectors like this, it could be ambiguous which field are we talking about, which field do the scalars come from. If the field is not specified, you may assume the field is the field of real numbers. Obviously, if there's the letter i present, clearly we're talking about the complex numbers. But if I just gave you a vector with some integers in it and you're like, oh, am I working mod 5 or am I working mod 7? You might not be able to tell the difference. Therefore, if you're going to be working in modular arithmetic, there will be some indicator. You'll actually see the symbol like z7, or you might see something like mod 7. There will be a clear indicator which field you're working in. If no indicator is mentioned there, assume you're working over the real numbers. So we can assume that these two vectors provided are vectors that live in R2. And so question number one, given these two vectors, you're asked to compute a linear combination involving these two vectors. So 2 times u plus 3 times v. We should know how does vector, how does vector addition, how is it computed, how a scalar multiplication is computed, what are the appropriate properties for linear combinations and such. So we first started doing linear combinations in section 1.3, turned to the homework to see some other examples of like that. Question number one, you'll be asked to compute some type of linear combination of vectors. Now it might not be over the real field. It could be over different vector spaces. So do be aware of that. Question number two, it's going to be asking you to recognize when a matrix is an echelon form. A echelon form was introduced at the end of section 1.5. And to be an echelon form, there was three conditions that have to be held. So first of all, the first condition is that if you have any rows of zeros, much like this matrix has right here, those rows of zeros must be at the bottom of the matrix. That's the first condition. The second condition is that your pivots must make a downward staircase. That is, so if you're looking at pivots, they have to be making this downward staircase down the matrix. Now that doesn't necessarily mean there has to be a pivot in every column. You could, for example, have pivots in the following situation. This would be an appropriate situation as well. And then the last condition is that when you do have a pivot, let's say we have pivots like this, these two pivots, we have to have zeros below the pivots to be an echelon form. And so with those three conditions, we can check whether a matrix is an echelon form or not. Echelon form is not the same thing as row reduced echelon form. Now I should mention that every matrix that's in row reduced echelon form is necessarily in echelon form, but echelon form is a broader criteria. And so for example, to be an RREF, in addition to be an echelon form, you must also require that the pivot positions be one. That's not required for a general echelon form. To be RREF, you do need the pivots to be one. And likewise, to be an RREF, you need the numbers above the pivot position to likewise be zero. Those last two conditions are not required to be an echelon form. So make sure you read this question carefully. This version on the practice test, it gives you four matrices and asks which of them are an echelon form. Some of them will be, some of them will not be, right? But on the test that you actually take, this could be changed to row reduced echelon form, which would change the answer potentially. Now you will be given four matrices. Not all of them will be an echelon form. Not all of them will be row reduced echelon form. Some of them might be, some of them might not be. Based upon the answer types, if you believe matrix A is the only matrix which is an echelon form, you would select then choice A. Same thing if you only believe B or if you only believe C or you only believe D. I'm also allowing the possibility that two of the matrices could be an echelon form or in row reduced echelon form. So if you think A and C is in, if that's echelon form, you would check this option that's allowing that. But of course, notice A and C is not an option means you made a mistake somewhere. It's like, oh, I meant to say A and B or something like that. So do choose the option based upon which ones you think are an echelon form or in row reduced echelon form. Now, I guess one last thing I want to say before I leave this question number two is that although a matrix could be put into echelon form, I'm talking about the current state of the matrix. Is the current A, is the current B, is the current C or D? Are those matrices in echelon form or REF? Not about the potential if they could be. Question number three is going to ask you to do a calculation about the vector matrix product as was introduced in section 2.2 about matrix equations. So you'll be given a matrix, you'll be given a vector, and then you're asked to compute the matrix vector product A times X. Now, although you could use a calculator on this one, this question's not going to ask necessarily for the final product, but it's asking for the intermediate step. The idea here is that a matrix vector product is defined to be a linear combination of the column vectors of the matrix A, where the coefficients in the linear combination come from the entries of X. So you'll see this intermediate step. You're recognizing that, oh, A times X is a linear combination of the column vectors of A. What combination is that? Well, the coefficients come from the vector X right there. So it's a fairly straightforward calculation. You'll see that on question number three. Of course, the matrix A and X could change. The dimensions of the matrix could of course also change. This is three by three. We could do three by two, two by three, four by five. Any of those would be appropriate. I'm mostly going to pick some matrix and vectors so that it fits on the page in a convenient way. So it's not going to be too huge. Question number four, we're returning to augmented matrices right here. And so we're going to be asked, I'm asking you to analyze based upon these augmented matrices. What can we say about the nature of the solution set? So these were things, this is a conversation we had at the beginning of section 1.6. If we have an augmented matrix, what can we say about it? So this particular question asked is the system consistent? So what can we see about these matrices to determine whether the corresponding linear system is consistent? Now, I'd be aware that some of these matrices, you should be able to identify whether it's consistent quite quickly, maybe because it's an echelon form or it's an even rubber-duced echelon form. But you can determine whether the system is consistent just from echelon forms. There's other ways of detecting consistency that don't necessarily involve doing lots of raw operations, although you're welcome to do that if you need to. This version of the question asked about consistency, you could be asked, does the solution have, which of these systems are inconsistent, that's a possibility. You could be asked, which of these solutions, which of these systems have unique solutions, which of these systems have multiple solutions. So I might be asking any of those things. So asking sort of about the nature of the solution set. So we did talk about the options right in 1.1. You could have an inconsistent system, there's no solution. You could have a unique solution. That would be because the equations in the system are linearly independent. That would also be equivalent to saying that the row vectors and the coefficient matrix are linearly independent. But you could also have multiple solutions. We call that the dependent case earlier. And that's because your equations in the linear system would be dependent if we view them as vectors or same thing. If you'd look at the rows of the augmented matrix, they would be linearly dependent. Now the dependent case in section 1.1, one thing I want to clarify is in 1.1 we said the dependent case meant there was infinite solutions, but that's only because the real numbers is an infinite set. I actually prefer the term multiple solutions because if you're looking at a finite set like Z5 or Z7, you don't get infinitely many solutions. You get multiple solutions, you're going to get multiples of the field size itself. So if you're having, for example, a field of size 5, Z5, multiple solutions mean that you have to have five solutions or 25 solutions or 125 solutions or 625 solutions, you know, looking at powers of five right there. So those are some things to look out for on this one. Again, more than one matrix could be consistent or dependent or inconsistent whatever it's being asked here. So if you think it's just matrix I, then you'll select choice A. If you think it's matrix I and I, double I, right, so one and two, then select choice D. If you think it's all of them, then you would select choice G, right? If you think it's none of them, none of the above is also a possibility. Don't just choose none of the above because you don't know what to do. Choose that only because you're convinced the three matrices are inconsistent in the situation or whatever the question is asked to be. The second page of the test in question number five, you're going to be asked to determine whether this, which of the following sets of vectors, S, T and U, which of them are literally independent, or maybe I'll ask which of them are linearly dependent. That same question comes up. This is a question type we did in 2.3. In particular, you had a homework question that was like quick, you know, determine which ones are literally independent in less than 10 seconds. There were some properties mentioned in section 2.3 about whether a set of vectors was literally independent or not. There's some things to check for, right? So for example, there's something about, if a set of vectors contains the zero vector, what does that say about independence? If you only had two vectors, there was a quick check right there. If you only had one vector, what happens when you have too many vectors? Notice we have four vectors in R3. There were some quick checks we could do. Question number five is going to ask you about these things. I mean, you could do a full blown row reduction to echelon form to determine independence, but this one is intended to be sort of a quick one. And like we've seen before, there are three sets provided S, T, and U. And based upon which ones you think are independent or dependent, make your choice. If you think only option S is independent, you'd select choice A. If you think S and T are independent, you would choose choice D. If you think S, T and U, well, then you'd be wrong, because that's not an option. So do make your choice accordingly. Question number six, also multiple choice. This is going to ask you to solve a linear equation over some field. So this was like some exercise that we did in section 1.2 about fields. Now, you're probably not going to be given the real numbers because that's too easy. So you'll probably be given either a finite field like working mod 11, mod 7, mod 13, it'll be a prime number. Or of course, you might be asked to solve it over the complex numbers. How do you solve a linear equation? Well, the idea is you're going to move this number to the other side and you have to then divide by the coefficient. But how you add its attract numbers, how you multiply and divide numbers does depend on the field. So you're going to be tested on some of these arithmetic calculations you've learned from that field from section 1.2. Question number seven is going to ask you to perform some row operations. And so like we learned about in section 1.5, there are the three elementary row operations. There's the replacement operation, there's the interchange operation, and then there's the scaling operation. And so as we row reduce matrices, we do combination sequences of those three operations. You'll be given a matrix right here as your prompt. In fact, I might give you this exact same matrix because knowing what the matrix is doesn't really change anything. It doesn't give you an advantage on this question. You'll be given this matrix and you'll be asked to perform two elementary row operations to that matrix. Now these elementary row operations will essentially be randomly assigned. I'm not necessarily giving you row operations that'll get you closer to echelon form. It's just like, hey, follow the instructions. In this example, you're supposed to replace row 1 with row 1 minus 3 times row 2. In terms of getting this to echelon form, this is actually kind of stepping the wrong direction. But then following that, you're then going to replace row 1 with row 1 plus 2 times row 3. So make sure you do the row operations in order. The order of operations can affect the outcome. And so some of these distractors might actually be if you switch up the row operations. So make sure you're paying attention to the row operations. You'll be given two or three row operations. Nothing too complicated. But be aware that if you were to like say, you interchange row 1s and 2, and then you add row 2 to row 4 or something like that, switching the rows changes who's row 2 is. And so when you look at when you look at the second row operation, you're replacing row 1 with row 1 plus 2 times row 3. You're talking about the current row 1 with the current row 3, which because of the previous operation, row 1 actually changed. So do make sure that you work those out accordingly. It should hopefully not be too difficult of a question. You're just asked to perform these row operations. And that then brings you to the end of the of the multiple choice section. Going to the next page, we then will start the free response section. And in this free response section, no, there's no blank line provided. You'll write your answer in the space. There's not really a lot of work to do on question number eight. What you'll be given will be something like the following. You're given maybe a system of linear equations like you see right here. And you're asked to express the linear system as a vector equation and a matrix equation. Vector equations, we talked about those in section 2.1. We talked about how they're equivalent to systems of linear equations. That equivalence is what I want you to express right here. And then likewise in section 2.2, we introduced the matrix equation, right? Ax equals b, which is the same thing, of course, as x1, a1. That's a poor looking plus, plus x2, a2, all the way down to xn, an is equal to b. So we have this vector form of this linear system. We have the matrix form. And I should mention also that we did learn about the augmented matrix, right? 1.6 is where we associated linear systems to augmented matrix. I take that back. I guess that was actually in 1.5 that we connected the system to the augmented matrix. So be aware that you'll be given one of these four objects. You'll be either given a linear system, you'll be given an augmented matrix, you'll be given a vector equation, or you'll be given a matrix equation. And so given one of those prompts, you'll then be asked to produce one of the two of the other three. So in this case, if I give you the linear system, I want the vector equation and the matrix equation. If you provide the augmented matrix, unfortunately that was not requested. And although it's related, be aware that the matrix equation is not the same thing as the augmented matrix. This linear system could be expressed as an augmented matrix, in which case you're asked to find what's the linear system, what's the vector equation, or what's the linear system, what's the matrix equation. Be aware that there are four ways of representing the same problem. You'll be given one and you're asked to translate it into two of the other forms, not all three of them. So do pay attention. If you don't provide the right one, that could be a problem. And the experience I have is that students often mix up the matrix equation with the augmented matrix. I've had many times where I asked for the matrix equation, but students provided the augmented matrix. And therefore the unfortunate loss of the points here because they didn't follow the instructions. So be aware of that one, not a lot of work to be shown here, just write the translations there. Question number nine, this one you'll be asked to, you know, based upon some of the things we've learned about in chapters one and chapter two, you'll be asked to, I mean, how do I explain this one? This is going to be more of a conceptual type question. It's not going to be like, here's something you memorized, regurgitate it kind of like the previous question. It's not like, here's a problem, compute the answer. It's going to be a more conceptual type problem. So this one, for example, you're given a statement and you have to declare whether it's true or false. A system with more unknowns than equations has at least one solution. So if you have more unknowns than equations, the system is necessarily consistent. That's the statement here. Is that a true statement? If it's a true statement, you should give some explanation on why it's true. What's your evidence that supports it? Now, if you think it's a false statement, you would select false, and then you would provide a counter example of why it's false. Like, oh, here's a, here's a system of equations, yada, yada, yada, yada, yada, for which it's inconsistent, you know, no solution, something like that. That's what you would do for a true false question. No, that's not the only type of question that's necessary here. I've asked questions, which were like, okay, if you have a, say, four by two matrix, okay, you have a four by two matrix, what are all the possible echelon forms of such a matrix? I've asked a question like that before, in which case, okay, if you're four by two, so you're gonna have four columns, so four rows, two columns, well, you could get like the zero matrix, right? The zero matrix is an echelon form. That's something that people don't realize. Because if you go through the definition of, and I shouldn't say, well, let's not do echelon form, let's do like RREF, that's a little bit better, because there are fewer options there. So people don't always realize that the zero matrix is actually an RREF, because if you go through the definitions, rows of zeros are always below rows, non, nonzero rows, which as there are no nonzero rows, there's no contradiction there. Great. The pivot positions make a downward staircase. Well, this matrix, the zero matrix doesn't have any pivot positions. So I see no counter example to the downward staircase. Great. Every number below a pivot position is zero. Well, as there are no pivot positions, there's no problem there. Remember, zeros being below a pivot only applies to pivot columns. If you have nonpivot columns, don't matter whatsoever. And so the zero matrix is an echelon form. Is it in RREF? Well, all the pivot positions are one. Well, I don't see any pivot positions that aren't one, right? So it turns out that statement's actually true. And then lastly, every number above a pivot position is zero. Again, as there's no pivot positions, there's no pivot columns, that criteria only applies to pivot columns. So you can see that because this matrix has no pivot columns, most of the conditions don't apply. So the zero matrix is an echelon form. It would be a four by two matrix. Another option would be something like you have a one right here, zero, zero, zero. The number after it could be anything. So I'm just going to kind of put a star right there. This would be a matrix that's in a row reduced echelon form, because again, this number could be anything you want. Sometimes I like to just put a little asterisk there that, okay, asterisk is just a variable. It could be whatever you want. But that would be an example of a row reduced echelon form. And then the other type would be one, zero, zero, one, zero, zero, zero, zero. And so for four by two matrices, those are the only, these three families are the only row reduced echelon forms. There's a zero matrix. There's this one where you have two pivots. And if you have a single pivot, you have this, I guess I take that back, there is one other possibility. Your first column could be zeros. And then you have a one in the second one right there. So that is also another possibility. And so these would be the four types of row reduced echelon forms. Admittedly, this one has some variability to it, but just leave a variable in the position that we can't specify. And that would be the case. And so this is what I mean by some type of like conceptual problem. We're not really computing anything. We're not really regurgitating something we memorized. What we're trying to do is take a principle, we're taking principles we've learned about linear systems and vectors and matrices and applying it to some context like answering the true false question or listing examples of echelon forms. These will be our two questions in the free response, excuse me, in the short response section. As we move to the free response section, question number 10, it's only worth eight points. You'll be given a linear system and you're asked to solve, said linear system. To solve the linear system, you should then construct the augmented matrix. So something like what is A augment B, then you will row reduce this to its row reduced echelon form. If you want to do Gauss-Jordan elimination, or you could just row reduce it to any echelon form. And you could then solve the system of equations by just Gaussian elimination. I'm okay with either Gaussian elimination or Gauss-Jordan or some hybrid of them, right? You don't necessarily have to go, you can use, guess what I'm trying to say You don't have to follow the algorithm to a T. You just want to be using elementary operations replacement, scaling and interchange to simplify the matrix either to echelon form or row reduced echelon form. And then using that echelon form or REF solve the system either by back substitution or if you're an REF, you already have it. Now be aware that to solve the system, if the system is inconsistent, you would then report there's no solution, but you need evidence that there's no solution. Basically, you would need something like, oh, I got the matrix to echelon form. And then in echelon form, I find the contradiction like I have a row of zeros that's equal to two or something, right? So that would be evidence that the system's inconsistent. I would need that evidence. You can't just say inconsistent and expect to get full credit, even if it's true. If your system has multiple solutions, I do need to see the general solution expressed, right? So if Z turned out to be a free variable, I do need to see something like, well, X equals two plus three Z, Y equals seven plus five Z, and then Z is just a free variable. If that's the case, then I need to see that the general solution looks like two plus three T, seven plus five T, and then T. I need to see this general vector mentioned. And it's probably a good idea to mention that this is the general solution. If there are, in fact, multiple solutions, do something like that. If there's a unique solution, then specify what that unique solution is. I also want to mention that in this, in this problem, you do need to show all the steps, particularly as you're row reducing your matrix. I want to see every single elementary row operation. When you do replacements, I want to see it and indicate which row operation you're doing. Like oftentimes, we've done this in our course, right? We're like, this is row three minus two times row one or something. Indicate which row operations you are doing. Be very clear so that I don't have to guess what you're doing because when your grader has to start guessing what you're doing, that probably means you're going to be getting some demerits. It should be clear what is going on there. So indicate interchanges. Indicate when you ever use scale or row. Indicate when you do replacements. Show me the steps. This is our first exam, and therefore this is your opportunity to prove that you can row reduce a matrix. So I need to see all of the steps associated to that. If you have a graphing calculator, I'm aware that you can throw your augmented matrix in a typical graphing calculator. You hit the RREF command and it'll give you the final answer. That's great, but I need to make sure that you could do it without the use of technology. Now, by all means, use your calculator to help you with arithmetic. I don't care about that. This is not linear arithmetic. This is linear algebra. I need to make sure you understand the algebraic stuff going on here. And so these, you're going to be using either Gaussian elimination or Gaussian elimination or something resemblance of those algorithms we learned. There's examples like this in section 1.6. Question number 11, this will be worth 10 points. This will be a question about linear transformations like we first learned about in section 1.4. You will be asked to prove whether the given formula provides a linear transformation or not. And be aware that to show that something's a linear transformation, I need two conditions. The first condition is you are going to show that if you have arbitrary vectors, so if you take vectors, say, say x plus y, you want to show that this is equal to t of x plus t of y. And be aware that what does it mean to take x plus y? Well, x and y themselves are probably the vectors. You have like an x1 plus a y1, an x2 plus a y2. You're going to plug these numbers into this formula, okay? And then this is going to equal on the right hand side. Now there's going to be some intermediate steps right here. This should look like t of x1, x2 plus t of y1, y2. Didn't quite give myself enough space there. So this is what you're trying to show. You're trying to show that t of x plus y is equal to t of x plus t of y. You had a homework question similar to this. There were some other examples in the textbook that were not assigned. Use those as examples. Show that this map preserves vector addition. But that's not just it. You have to also show, to be a linear transformation, you have to also show that t of cx is equal to c of tx, like so. This question, if you understand this proof template, it's fairly straightforward. You're going to have an equation right here. Use that equation and you can rip it apart to show that this map preserves the vector addition and skill level modification. If you look at the solutions that are in the textbook, compare this with the example we did in our lectures, then you're going to see that this is a template, a proof template. If you follow it, you'll be able to get this one right. So I don't want you to get overwhelmed when you see things like proof. Yeah, mathematical proofs can be very difficult at times, but this proof comes with a template. You change the appropriate parts, you follow the template, and you'll get the correct proof each and every time. Question number 12. This question is the least amount of the free response section. And that's because it's only worth five points. This will be like some of the problems we saw in section one. Sorry, that's 2.4 on affine geometry. You'll be given some data about an affine set of some kind. So for example, we have a plane in R3. It passes through a given point, negative 211, and we know it's parallel to two provided vectors. And so we want to come up with the vector equation. It's a plane, so it should look like x equals x0 plus su plus tv. So specify what's x0, what's u, what's v, and then also provide the parametric equations. x equals yada, yada, yada, y equals something, z equals something, or if you want to call them x1, x2, x3, whatever you call the variables. This is what we're looking for on this one. And again, these questions aren't too computational difficult. With this one, we can actually jump to the vector equation. The parametric equation is pretty quickly, but a variation of this is you might be given three points on the plane, and therefore you have to take the differences. You know, you pick x0, and you subtract it from the other two to get the spanners, and you go from there. So other than a little bit of subtraction, there's not a lot of arithmetic on this question, which is why it's only worth five points. It really comes down to understanding the general formula for a flat. Question number 13, you'll be given a set of vectors, and that set of vectors could be represented as a matrix, right? A matrix essentially is a set of vectors where we think of the column vectors of the matrix as the vectors in our set. So given this matrix, which thinking of it as a set of vectors, we then want to determine, and you do need to provide the details of this, is this matrix, are the column vectors here literally independent? Are they literally dependent? And it does say provide proof right here. What that means for us is if you have a set of vectors, you're going to squish them together as a matrix. Oh, lookie there, it's already done for us. Then you're going to take that matrix and you're going to reduce it showing your steps step by step by step as you reduce the matrix, you're going to get it to some echelon form on the road, which echelon form is great, but you can save yourself time and effort by stopping at any echelon form. Once you get it to echelon form, you're going to look at the pivot positions. If there's a pivot. So let me make mention of this. If you have a pivot in each column, then that translates to mean that your set of vectors is linearly independent. If you have, if there exists some non-pivot column, though, if any column is lacking a pivot, then that translates to us, meaning that the set of vectors is linearly dependent. And that's what we have to show. So even though this asks us to show with proof, just like the linear transformation question we did a little bit ago, this proof is a template that we can do the exact same things each time as long as we change the appropriate parts. We take our set of vectors and we represent it as a matrix. We reduce that matrix to echelon form. And then once it's an echelon form, we can identify the pivot columns and the non-pivot columns. If any column doesn't have a pivot, then that means the vector set was linearly dependent. If every column has a pivot, then the vectors must have been linearly independent. There's no way of expressing one vector as a linear combination of the others. And that's all one has to check. I often say that when in doubt, row reduced. This is exactly a situation of that. If you reduce the matrix, the information from the reduced echelon form or just any echelon form will give us the information we need to solve this one. So it's just about determining whether independent or dependent, like we did in section 2.3, follow the template and you'll be okay. Question number 14, you'll be given a set of vectors, which it could be given as a matrix or in this case, it's actually listed as the separate vectors. Be aware that the two distinctions are really just fictitious, right? A matrix we're just thinking as a set of vectors right here. So if you have a set of vectors and then you're given some other vector, you're asked to determine with, again, with proof, right? This word seems a little bit scary, but it will follow a template that we'll see in just a moment. We have to determine whether the vector B belongs to the span of the set S or not or another way of writing this. Let's say that your set of vectors was given as A equals, you know, A1, A2, A3, A4. So let's say it's the exact same four vectors. This question could be alternatively expressed as is B inside of the column space of A, right? It's the same. It's the exact same question because the column space is just the span of the column vectors of the matrix. It's the exact same language right here. So we have to determine whether B belongs to this set, the span of vectors or not. Remember, the span is going to be the set of linear combinations of these vectors right here. And so this means we have to come down, we take the augment to matrix A, augment B, where A, it has as its coefficients, sorry, this matrix A, it's a coefficient matrix, it has as its column vectors, the vectors inside the set S right here. You then augment it with the vector in question and you're going to re-reduce this matrix to echelon form, re-reduce echelon form would be great, not necessary per se, but you're going to re-reduce that to echelon form. If this system is consistent, if this system turns out to be consistent, which we can identify consistency from any echelon form, that means that the answer would be yes. That is B does belong to the span of S. On the other hand, if this system turns out to be inconsistent, if it turns out to be inconsistent, then the answer would be no, B does not belong to the span of these vectors right here. So that's the distinction you have to do. Now in the case that B does belong to the span, that means B, if it's in the span of S, that means it can be expressed as a linear combination of the vectors from S and they want an explicit linear combination, that will come down from the solution to the system. The solution to the system will be the coefficients you slap in front of these things. So your X1 you find will be the coefficient of A1, the X2 you find will be the coefficient of that one. The X3 you find will be the solution will be the coefficient of that one and then you add to that X4 times whatever you get for X1, X2, X3, X4. You solve the system. There could be free variables. If you have any free variables, I'd probably set the free variables equal to zero just to make life a little bit easier for you. And that's what you're trying to do. So you might have to do more than just say yes or no. And so this is a problem we saw in section 2.1 when we first considered whether a vector is part of a span or not. In section 2.2 we started talking about whether a vector belong to a column space or not, which is the same problem. And then we revisited the column space a little bit later, but those are the sections to go look for. And then question 15, this is the last question except for of course question 16, which is a reminder to turn in your notes when you submit your exam. Question number 15 is going to be asking you to look for counter examples to subspaces and to vector spaces. So in section 1.3 when we introduced the vector space, the notion of a vector space, you were given some homework questions where you're supposed to provide a counter example on why the given operations do not form a vector space. Like in the homework, we modified skill and multiplication. We took this hypothetical skill and multiplication and argued this skill and multiplication doesn't work for a vector space because it violates the distributive law or it violates the identity principle or something like that. Likewise in section 2.5, you had some homework questions which were given subsets of R2 or R3 and you're asked something like that and you're asked to them prove why was this set not a subspace. Now to not be a subspace is yet to argue does it contain the zero vector? Is it closed under addition? Is it closed under skill and multiplication? And so you have to provide a counter example on why that's not the case. So this example will hopefully look familiar to you. Let Q be the first quadrant in the XY plane aka R2, right? So we want entries which are non-negative prove that this is not a subspace. This is exactly the example we did in our lecture. And so I want to provide an answer very similar to that. So you can prepare this by looking at examples from the homework from from the lectures. You'll be asked to prove something similar to that. And when I say kind of prove, I really mean like you're providing a counter example. So in this case, again, you're asked to show something. Oh no, you have to prove everything on this test. All of these are templates, right? If I'm trying to show that this set is not a subspace of R2, I would go through the axioms. Does it contain the zero vector? Yes or no, make a statement. If the answer is yes, I don't write it down. I move on to the next one. If the answer is no, I provide a counter example and then we're done. Then you look at the second one. Is it closed under addition? I think about it. I consider it. If the answer turns out to yes, it is closed. Then I move on to the next one. I write nothing down. And then if it's if the answer is no, it's not closed under addition. I provide a counter example. You look at the third axiom. Of course, if ever the answer is no, you can be done and move on to something else if you skipped a question. Then you look at the last one. Is it closed under skill or multiplication? If the answer is yes, then it means you actually missed you made a mistake. Go back and check the other ones. If the answer is no, then I provide a counter example. Once you find a counter example, you're done. You only have to disprove one of the axioms. You don't have to prove any of the axioms and you do not have to disprove multiple axioms. You just need one axiom to disprove. That gets us to the end of this exam right here. Like I said, question number 16 is going to ask you to submit your notes for this exam. So there we have it. That's our exam here. It's, you know, you want to do well on your exam, but in semesters past students have done really well on these exams. If you study and you've been doing your homework, you've been engaged in class, then I don't have a lot of worries for you whatsoever. I do. Of course, if you have if you've been skipping class or you haven't been doing your homework, then of course I do have some concerns, but this this test is for the most part very mechanical, even on questions, which are like prove this, if you've those proofs follow a certain template, if you've been doing the homework, you're probably familiar with that template. And by all means study some more before you take this exam, right? The exam will be available for some multiple days. Take it when you're ready. I also wanted to mention that this this practice exam does have the solutions provided right here. So you can check your work and you could actually take this as a practice exam. I'm not going to go through the solutions of this one by one right now. There are some videos for solutions that you can find on Canvas. Take a look at those when you are ready. I recommend you take the test, you know, work through the questions yourself before you try to check the answers. I think it'll be more beneficial for you, but those solution videos will be provided to you and you can look for them on Canvas. Have a have a great day, everyone. Best of luck on the exam. If you have any questions, do not hesitate to ask me. I want to help you as much as I can. Just let me know. And we can talk about this test at length. Of course, during the testing window, as we won't necessarily all be taken at the same time, I do have to mention that you cannot talk to other people about what's actually on the test based upon what you've seen. Like if let me clarify, right? If you're working with a classmate or in a study group, you're like, I don't understand how to show that a linear transformation is in fact linear, right? How do you prove that? Of course, you can work together and talk about how you prove that. But if you're like, oh, on the test, the question was t of x1, x2 is equal to 3x1 plus 5x2. You know, that type of thing, we're actually talking about specific questions on the test because like someone saw the question, those type of conversations would be considered cheating, academic dishonesty, and those will not be tolerated whatsoever. Please, please, please don't do anything like that. Make sure your work is your own effort that your answers are your answers, your own answers, not influenced by anyone else. If you do have questions, I would love to talk to you, work with your classmates, work with me. That's perfectly acceptable. But during the testing, the examination window, please don't talk about the test because no one wants, no one wants to send you to have a conversation with the dean of students. I don't want that, you don't want that. But of course, people who violate the student, the student code of conduct and particularly with academic dishonesty, I have no choice but of course, to act in accordance to the policy there. That's sort of a bummer way to end this video. I apologize for that. But let's try to state on a happy note. I've had, I've used this similar type of exam, not the exact same questions, of course, but I've used this type of exam for many, many semesters. And for the vast majority of students, they do exceptionally well on this exam. All of the exams, you are in linear algebra right now because you know how to study mathematics, you know how to succeed in mathematics. And therefore, trusting yourself, believing yourself, put in the work, right? That might mean going back and finishing homework assignments you didn't finish. It's better to do it late than it is to do it never. Study, take the time to do it, ask questions, talk to tutors, talk to classmates, start discussions with classmates, come talk to me during office hours, or try to seek me out at some other appointment or drop by or whatever is appropriate. Let me know what your questions are and I want to answer them. And if you do, if you put in that effort, I anticipate high marks on this exam. I wouldn't be surprised if the class average was anywhere in the 90s that's happened before, but it will require the work on your part. You can do it, I believe in you, please believe in yourself. And I'll talk to you next time. Bye, everyone.