 It's time again for an exam in math 2270 linear algebra For students at Southern Utah University in this video I want to review the topics that will be covered on exam 2 now This is the second exam in this course And so the structure and policies for the exam are basically going to be the same as you saw on the first exam So I don't necessarily want to repeat all of those details Also certain details about time place or manner of the exam change from semester to semester So look for that information either on canvas or reach out to your instructor as there's probably announcements relative to that So things about like what notes can I use what calculator? Can I use again consult that information on canvas in this video? We're just going to talk about the topics which are on this exam So exam 2 is going to pick up where exam 1 left off So to remember here exam 1 went from the beginning of the book until 2.5, which was about subspaces Our exam that in the current conversation exam 2 we're going to pick up where we left off We're going to go from 2.6, which was about solution sets to systems of linear equations the general solution and we're going to go up to section 4.3 which is the start of chapter 4 about orthogonality and so The idea of projections Will come up on this on this test right here So that's the topics that you want to prepare for for this exam in terms of questions There'll be three types of questions on this exam multiple choice sure response and free response The multiple choice as the name suggests you'll be given multiple options You select the correct answer and you'll get full credit for selecting the correct answer and no other answers You get no credit if you select nothing or if you select an incorrect choice no partial credit is available for multiple choice It's either all or nothing based upon that response on this exam There will be three multiple choice questions worth five points each The next category is the short response questions, which the short response questions You will not be given choices like the multiple choice The questions will be relatively simple I would say very much like a multiple choice question but That in the short response you just have to write down your answer You don't necessarily have to provide any supporting work I mean if some of these questions might require that there might be work to do but you do not have to supply You only have to supply the final answer for full credit And so full credit can be given for that final response Although because these questions might be a little bit more involved You could potentially get some partial credit if your answer is if your final answer is partially correct Or if you have supporting work that leads to an incorrect response So it's a good idea to show your work There will be space provided on the test packet to do so But it is not a requirement on the show response section There will be five questions in that category and there should and those are gonna be worth six points each Just a little bit more points than a multiple choice The remaining questions for which there will be 15 questions total The remaining questions are what we call the free response section in the free response section You do have to show all of your work for full credit Just including your final answer will give you little to maybe no points whatsoever You need to show all your work and that might mean showing Things like row operations and things like that So let me then show you the specifics of these question types So you get an idea of what to study for what to expect on this exam So question number one As it's currently phrased, it tells you, you know, suppose you have a matrix A Which is two by three and a matrix B, which is three by five What would be the shape of A times B? That is, what you know, what are the dimensions of that matrix? Is it going to be seven by five three by two seven by 11? So this has to do with matrix multiplication These are the types of things we did in section 3.1 When we introduced the matrix operations that chapter three is all about So matrix multiplication was definitely the most important one But we could talk about scalar multiplication, matrix addition, transposes traces And so this question will ask you to do sort of a partial calculation It doesn't have you do all the details But, you know, if you like add together some matrices and multiply them and transpose them Could you predict what the dimensions of, you know, what the size, the shape of the Resulting matrix would be If you take inverses, how does that affect the shape of the matrix? So be prepared for something like that And I should always remind you that when you see this practice test This practice test first of all is not collected The solutions to this can be found at the end of this document I'll also provide some solution videos that you can find on this channel here But the thing I really want you to be aware of with this practice test Is that if you see a question on this test That does not mean that that question you'll be on your test And if you're something missing on this practice test It doesn't mean that question couldn't be on your test This practice test is just a sample of what you could see It should supplement your studying And it shouldn't be your entirety of your studying here And so that gives us a discussion of question number one Expect to do some type of matrix operations But you don't necessarily do the whole calculations More conceptually what size, what shape would the matrix have at the end Question number two Is going to be a question about inner products in some capacity So you might be asked to compute an inner product That could be the dot product for real vectors That could be the Hermitian product for complex matrices Complex vectors, excuse me So you have to watch out for that thing For the complex, you have to make sure you take the appropriate conjugates If you don't take conjugates, you're going to get a calculation wrong That's probably the most common mistake there So question number two could involve a calculation Involving the inner product, the dot product or Hermitian product It could involve a calculation of the outer product When you put two matrices together to form a matrix It could involve like in this question right here The length, the norm of the vector You might be asked to compute the distance between two vectors Again, it's going to be some fundamental calculation Very algebraic that involves the inner or outer product In some regard or another Now this could also be in combination with other vector matrix operations So you'll see here, for example, that we have linear combinations Addition of vectors, scalar multiplication There could be matrix multiplication also in play Very similar to Homer questions you saw in section 4.1 and 4.3 Question number three You'll be asked to identify which of the following sets is orthogonal And so this is very similar to a question we saw on test number one Which we're given three sets and had identified which sets were linearly independent We have to decide which sets are orthogonal And so for the most part, that means checking case by case Are these orthogonal? Are these orthogonal? Are these orthogonal? If you ever find an inner product that didn't turn out to be zero That means that the set is actually not orthogonal Now some things to watch out for here That if any set contains the zero vector Then that set is automatically not orthogonal Now this is a little bit confusing sometimes Because the zero vector does have the property that it's orthogonal to every vector So the inclusion of zero doesn't actually frustrate Checking whether you have orthogonal pairs But by definition orthogonal sets are not are necessarily non-zero This was to ensure that orthogonal sets were always linearly independent So if the zero vector is contained in there It is not an orthogonal set automatically Another thing to watch out for Is if you have just a single a single vector inside your Your set right here like let's just take the vector one two three A single vector is actually considered an orthogonal set So long as it's not the zero vector Because like I said the zero vector cannot be included inside of orthogonal sets But otherwise if you just have a single vector This is actually considered an orthogonal set And the reason for that is that there are no pairs of vectors Which are not orthogonal because there are no pairs of vectors It's sort of vacuously true Also in that vein if you ever see the empty set This is also considered an orthogonal set Because it doesn't contain the zero vector So everything's non-zero And there are no pairs of vectors which are not orthogonal So it's orthogonal And so I want you to be very careful about that Because this could be a confusing thing for students But these are two important examples I want to throw out That oftentimes have confused past students So be aware of such things If you need to study more about orthogonal sets Orthogonal vectors as well as orthogonal sets Were introduced in section 4.2 Of our textbook linear algebra done openly And that's going to be the multiple choice section Those three questions right there So very simple things Like in the homework for 4.2 There were some questions that were like Quick, decide if these things are orthogonal No, I think there was a question like that Well, it's going to be similar to those Very basic calculations Is this set orthogonal or not All right So the short response section It's a little bit longer than multiple choice Let's look at some of the questions here So remember all of these questions are worth six points each The first question number 4 that you're going to see Is you're going to be given a linear transformation You don't have to prove that to linear transformation Like we did on test number 1 You'll just be given a transformation And you may assume it's linear What you are asked to do is to construct The standard matrix representation Of that linear transformation Every linear transformation can be represented by a matrix What is that matrix going to be? Well, this standard matrix always will have the format Well, the first column is just evaluate The matrix, or evaluate the transformation at the first The first vector, the first column of the identity matrix This vector E1 So it has a 1 in the first position Zero everywhere else The second column will be evaluate The transformation at the second Column of the identity And you'll go all the way down Until you get T of E N right here So we just have to, you know Like with this one right here The idea is if I set X1 to be 1 And X2 to be 0 What do I get? That gives me the first column If I set X2 to be 1 And X1 to be 0 All the other variables to be 0 That's going to be my second column That's how we compute these things And this matrix representation We saw at the end of chapter 3 So you can see some more examples Like this on in section 3.7 That's going to be question number 4 Question number 5 This is sort of a curious little question And there's a very good reason Why it's in the short response section So you'll be given a matrix A It'll probably be 3 by 3 real numbers Don't want to make it too complicated for you But this matrix A You're going to then see a below it A sequence of row operations That convert the matrix A into the identity So A is row equivalent to the identity That means it's a non-singular matrix Well, since it's a non-singular matrix That means it can be factored As a product of elementary matrices Now, if you recall how we did this Elementary matrix factorizations came from factor Well, it came from row reducing the matrix To the identity, right? And so we could each elementary row operation We do corresponds to a factor Of the elementary factor Well, every elementary row operation Corresponds to an elementary matrix As part of this factor So as you look at this matrix You're going to look at the sequence of matrices You just see it step by step What are the operations in play here? So if you compare like the first matrix To the second matrix It looks like there was an interchange Going on right here Then as we go from this matrix to that matrix It looks like there was some type of row replacement Right, this 4 went to a 0 So it took probably like row 3 Minus 4 times row 1 or whatever And so each of those Each of these steps There's going to be some row operation That goes there So what's the corresponding elementary factorization? So to form that factorization A equals You're going to get a matrix For each of the operations So I see like 1 operation 2 3 4 and 5 and 6 So it looks like I should have like 6 matrices When I'm done here I should have drawn these a little bit smaller But you get the idea And so then the first matrix Is going to be the first step It's going to be the first step But you're going to be You're going to have to take its inverse, right? So the inverse operation Well, if you had it If you had interchange The inverse of interchange is just interchange Then you have the second operation But you have to take its inverse as well So in that situation If we took You know, if I think that's like Your place row 3 With row 3 minus 4 times row 1 Then actually the corresponding operation Is going to be row 3 plus 4 times row 1 And then write the corresponding elementary matrix In that situation All right And so we talked about elementary matrices In section 3.4 3.1 introduced matrix operations 3.2 we talked about properties Of those matrix operations 3.3 introduced the idea of non-singular matrices And their inverses And then 3.4 we introduced the elementary matrix As a tool That eventually led to the inverse algorithm That we then have a way of computing Inverse matrices So in question number 5 You will simply write down the elementary matrix Because you have the sequence of row operations In front of you You just have to translate the sequence of row operations Into elementary matrices Make sure you go in the correct order And make sure you take inverses there Those are the two Those are the two major things That students make mistakes on this one They might go in the wrong order That's usually okay Because we're going to go left to right on this one But they forget to do the inverses So you have to know how to translate A row operation into an elementary matrix And make sure you take the inverse of that Just like we did in section 3.4 So there's really not a lot of work to show on question number 5 It just needs to make sure the factorization is correct Question number 6 is going to be very similar to that But in question number 6 We're going to focus on the LU factorization Like we did in section 3.5 All right So with the LU factorization Again, it's very similar to the elementary factorization That you're going to see a sequence of row operations Although I might Just to make sure I fit it all on the screen I might do more than one operation at a time Which is fine You should hopefully be able to see Like sometimes you could do two replacements at the same time Because the order doesn't really matter in that situation Not a big deal whatsoever On this one, you're going to come up with the LU factorization So you need a matrix L and you need a matrix U L will be a square matrix Which is unit lower triangular It'll have the same number of rows and columns As A has rows Now if A is a square matrix like you see in this example That makes life a little bit easier U will be a matrix in echelon form And it actually should be the first echelon form you find When you go down the sequence of row operations To form L, we can only use lower Replacement operations So that L becomes a lower triangular matrix Very similar to what we did in on number 6 But we practiced this LU factorization So I just need to see L I just need to see U So U Be aware that with this sequence of row operations It goes all the way down to the identity Can't see it on the screen right now But this sequence will go all the way down to the identity So we're kind of playing where's Waldo a little bit U is in there somewhere, right? You have to decide one of these matrices is going to be U And then once you decide on U You're going to squish together the previous matrices Basically, I should say the previous row operations To create the matrix L So give me the LU factorization If you see the sequence of row operations there Question number 7 is going to be very similar to these ones as well And question number 7 You'll be given the factorization into elementary matrices For the most part You'll be given a factorization of A Which will be a 2x2 matrix And then you're asked to describe what are the geometric consequences What is the effect geometrically of these matrices? So what does A do to the plane? And so you would be going from this You'd be looking at this factorization Now unlike questions 5 and 6 Number 5 and number 6 You're going to go left to right But you take inverses On question number 7 In order to interpret this correctly Because the vector The vector is actually going to be sitting over here X and Y It sits on the right So the first matrix in play is actually the one on the right The next one in play is the second one And then the third one And then keep on going And so because of that We want to move We want to work right to left Now you're going to list these things You know okay What does this matrix do? What does this matrix do? What does this matrix do? What does this do? What does this do in terms of Shearing maps Stretches Compressions Reflections Rotations For projections Those are the types of things That I want to see here And the hardest thing about this one Is you really make sure you go right to left There is no worry about having to take inverse operations whatsoever Make sure you go right to left That's the most common mistake So we talked about these geometric transformations Mostly in section 3.6 But the idea of a projection was introduced later on In section 4.3 So also be prepared for that Like the example you see in front of you also It has a projection in there So you should be able to accommodate for that And so then the last The last of the shore response questions Is you'll be given a matrix equation And you'll be asked to solve for the variable matrix x So solve for x This is like what we did in section What is that one? 3.2 Nope 3.3 excuse me Because that's inverses in play 3.3 in the textbook Now when you're solving this You can assume that matrices are non-singular So I don't want you to have to worry about Does the inverse of a matrix exist or not Assume that when necessary There always is an inverse of a matrix So things are in by in and non-singular Just to make life a little bit easier for us But do be cautious that We are not assuming matrix multiplication is commutative We don't know that here So that this has consequences right So a times b does not necessarily equal b times a I would check as I'm grading this To make sure that when you move things From one side of the equation Is it a left multiply or is it a right multiply? There's a difference there If transposes are in play Remember the transpose is a shoe sock operator It switches the order when you distribute it Inversion also does the same thing If you have a b inverse This is a b inverse Sorry a b inverse will be b inverse times a inverse So the biggest concern with question number eight Is does the student recognize that we have Non-commutative operations in play And I want to make sure that that We don't accidentally commute when we shouldn't have And so that will finish the short response section Let's move to the final category of free response Again these are ones where you have to show your steps You probably have access to a calculator Which can help you out here But you have to make sure you show sufficient work The final answers is not enough Like on question nine for example You'll be given two matrices And you'll be asked to compute for example the product This is the test I should say to assess That you understand the mechanics of matrix multiplication Multiply the two matrices together And you do need to show all steps A final answer will give you no credit on this one I again most many of you will probably Have a graphing calculator when you take this test In which case you can throw this in a graphing calculator And it will give you the final answer no problem I need to make sure that you know how to do this And so I need to see all the steps It's only worth six points But if I just get a final answer with no work You will get zero points on this question And of course the question could be a little bit more involved Than just multiply a times b There could be a few other matrix operations in play Like transposes, traces, addition of matrices So this would be a question really from section 3.1 About operations of matrices But you definitely will be asked to do matrix multiplication If nothing else you will be asked to do matrix multiplication That is the most complicated of all the matrix operations If I dare call it complicated But you know that easy versus hard is always a subjective matter You're trying to prove to the grader here That you understand matrix multiplication That's what you'll see on question number 9 Question number 10 This goes back to the start of this unit Section 2.6 in particular You'll be asked to provide the general solution To the non-homogeneous system a x equals b So what we're trying to see here Is that the general solution x This is going to look like some particular solution x naught Plus the general solution to the Homogeneous system So you get some combinations of ci, vi So this right here is the solution to the homogeneous system Basically I want to see Essentially the basis for the null space That's basically what we need to see here Although as a general combination A general linear combination And then you should have some particular solution in play right here Now the good news about this question right here Is that in the current form The matrix a is actually given to you in row reduced echelon form So solving it turns out to be really quick and easy And therefore it's just about kind of ripping this thing apart Now This question was with seven points On the test if this matrix is not in row reduced echelon form I do not need you to show me all the steps of row reduction If you start off with a augment b And then you row reduce this to be something like Let's say it's the identity augment It probably won't be the identity so that it has multiple solutions But you get some echelon form and then say you get like a What's called y whatever It needs a name So you know if you read your row and do this I don't need to see all of the row reductions If you just use that as a calculator to do that That's fine If you just you don't scratch paper and didn't put on the test I'm okay with that I'm not going to grade you on question number 10 about your row operations But I do need to see the echelon form That's necessary And then I need to see how do you extrapolate The general solution for this non-homody system from the echelon form That's what will be graded on this question number 10 Question number 11 This is a fun one I like this one a lot For which this question is going to ask you about Two of the three fundamental spaces of a matrix that we know so far So you'll be given a matrix a and it can be some big thing like this one So this one's three by five matrix You'll be given the original matrix a And you'll be given its echelon form It's row reduced echelon form So you don't have to worry about computing the row operations Because it'll be done for you You will be given a and you'll be given its RREF Now from that you'll be asked two things The first thing is a guarantee That you'll be asked to construct the null space of the matrix Well that is I want you to find a basis for the null space of the matrix And so that comes down to identifying pivot non-pivot positions And then pulling out information about the free variables in that situation So you need to be able to solve for the You need to be able to compute the null space Be able to find a basis for the null space of the matrix Now be aware that with the null space we introduced this back in chapter two Let's see what is the corresponding section It would be two point oh boy I'm blanking right now It would be two point six I think no two point seven Well when we talked about we could two point sevens when we talked about a basis and dimension We talked about how in that's in that section how you can find a basis for the null space So that's something you would definitely be asked to compute Now what else could you be asked to compute? So one question like you see on the screen right now Is you might be asked to compute a basis for the row space of a The row space of a was introduced into section three points two And you were taught how to construct a basis for the row space of a That had to do with finding the pivot positions the pivot rows and the echelon form And grabbing the pivot rows of the echelon form gives you a basis for the row space You could also be asked to compute a basis for the column space of a which also was talked about in section two point seven And remember they're finding a basis for the column space They came down to finding the pivot columns and then finding the corresponding Columns in the original natrix a And so you'll be asked to compute you will be asked to find a basis for the null space And then maybe the row space or column space I won't ask you both because they're both fairly easy. There's no computation there It's just something you read from the rif but you should be prepared for either one It's kind of random. That's going to happen there Question number 12, remember there's 15 short or free response questions total Question number 12 is going to be about computing inverse matrices So this will require the inversion algorithm that we learned in section 3.4 I'm not going to ask you for a two by two matrix. This is the algorithm where you're going to take Um a augment the identity and then you will reduce this to The identity augment a inverse So some things I need to see here is that do not stop here. This is not the finish line Uh, this will just give you matrix. I actually need you to tell me that a inverse equals a specific matrix If you just end with this augmented matrix, how do I know that you know what the inverse matrix is? You need to be explicit tell me the inverse matrix is this So just go the extra step and write that down here so that I know That's something I look for you don't want to lose a point just by that silly mistake And also as this calculation is primarily just row reduction I am going to ask that you show all of the row operations here I want to see the details on how you go from here to here. I need to see those without that I can't give you full credit on this one That's going to be question number 12 uh question number 13 You can see right here uh question number 13 is going to have to do with uh coordinates of some kind like we talked about in section 2.8 the last section of chapter 2 you'll be given a some set of vectors right here And so this is going to be a basis for a vector space. So in this case, this is a basis for r4 It's a non-standard basis for r4 Then you're going to be asked to compute You're going to be asked to compute the coordinate vector of The vector given with respect to these coordinates right here So the idea there is you're going to take your basis and you're going to augment the vector in question right here You're going to row reduce it You're going to row reduce it you'll get the identity here and then you're going to get the coordinate vector Right here. Just like the last question. You cannot stop here You need to actually tell me that oh the coordinate vector with respect to b coordinates is whatever it turned out to be Right now you have to be cautious on this one Because of the nature of these things you could actually get rows of zeros on the bottom That is a possibility if you can row some zeros just ignore those Do not put those into the coordinate vector, right? Don't do that That's that's a common mistake here. And again, just like question number 12 The vast majority of this question is Set up an augment to matrix and row reduce it and there and then interpret the ref Right. So just like question or 12. I do want to see all of the row operations on this question Because that's essentially the calculation. That's the showing the work part All right question number 14. This is the last page question number 14 Is very similar to question number 13 and that this will come from two points 2.8. Excuse me about change of coordinates It's actually I'm going to ask you to compute the change of basis matrix So you have a basis and a basis So we have two bases in this is a case for r3 and we want to compute the change of basis matrix So there's going to be two parts of this question. So to compute the change of basis Right following the order you see right here c to b you're going to write b on the left augment c on the right Then you're going to row reduce this you should get the identity of some kind in this case would be i3 And then you're going to get your change of basis matrix Right here Just like the previous two questions two things to mention that you do need to show all the row operations I do want to see all the details here, but you can use your calculator to verify it Do not stop here. You do need to be explicit. What is the change of basis matrix? You cannot just assume that I know that you know something right That I know that the change of basis matrix can be found right here But you need to tell me that is the change of basis matrix tell me exactly what it is So that's going to be very similar to the inversion algorithm. We saw on question number 12 But this one has an extra step You're then going to be given a vector in one coordinate system You're asked to change it to the other coordinate system. And so be aware that x in b coordinates is going to equal the change of basis matrix as you switch From c to b when you times it by c coordinates So just a very quick calculation once you're going to multiply a matrix by a vector nothing too complicated But don't forget that step you need to know how to switch from c coordinates to b coordinates If you have the change of basis matrix All right, and so then the last question question number 15 right here This comes from section 4.2 And this is going to be a question about the orthogonal complement of a matrix of a subspace. Excuse me You'll be given a spanning set for a subspace. So in this case you have vectors u v and w And you'll be given the spans which this is a spanning set I'm not guaranteeing it's a basis. Although it probably is a basis in this situation You do have a basis these three vectors u v u v w are limited independent But you don't need that We see that w is a subspace of r5 and you're asked to find your orthogonal complement The key observation here is that if you take the row space of a matrix a and you take it to orthogonal Compliment that gives you the null space Of that same matrix. So what you need to do is you need to construct a matrix a Right, you need to construct a matrix a so that the row space of a is equal to w And the easiest way to do that is just to put your vectors u v w to be the rows of the matrix a So then once you have a so that the row space of a is equal to w then you're going to row reduce this To get your ref Which do you need to show your steps here? Probably not It doesn't say specifically show your steps and there's much more to this problem than just row reduction So if you just if you don't show me any details r ref, that's okay Once you then have the ref right then you need to compute since the orthogonal complement of Since since the orthogonal complement of the row space is the null space then you need to find a basis Find a basis for The null space of a And then that will be the orthogonal complement w prime So that's what we need to do in this situation. So the orthogonal complement But we'll because you're trying to find a basis through a orthogonal complement That'll just be finding the basis of the null space of a if we chose a correctly So that's what we have to do you have to set up the matrix a row reduce it then find a basis And so I want to point out a couple things this question number 15 Requires essentially that you find the nulls the basis for the null space of a matrix previously We saw in question Number oh where did it go in question? Oh, I can't remember it was on the previous page question number 13. That's what it was on question over 13 You have to find the basis for a null space plus another basis, right? And then also on question number Oh boy, I'm gonna have to stretch myself on this one. It was the first question question number nine I think was what what it was. Uh, I guess we can always go back Whoops too far Question number nine. It was the first one from the from the free response Question number 10. I was off uh question number 10 Right here, we have to solve for the general solution to a non homogenous system That essentially requires you find the basis of a null space So if there's any if there's one thing that you need to know how to do At for this test is you need to know how to multiply matrices And you need to know how to find the basis of a null space if you do not know those two things You will fail this exam. Um, you also of course need to know how to row or do some matrix But that's something we tested on the first exam. So I'm assuming by now We're pretty good at that. But those three skills will be critical for this exam The last question there is technically question number 16 But that question is just to submit the notes That you were allowed for this exam for two points. That's just a pass fail thing You should prepare some notes prior to taking this exam. And so that'll then be question that that's that's exam number two Uh, so I think I think this is a pretty good coverage of topics study put in the work and you'll do just fine Of course, if you have any questions, feel free to reach out to your instructor Uh, that would be me and I'll be glad to help you through this process Just let me know if you have any questions