Birne Binegar: Okay, so I guess, we can begin so so first thing I want to do today is not set in stone date of the first exam I told you.
Birne Binegar: Earlier this week that Friday week from today would be sort of an optimal date.
Birne Binegar: So yeah any objections to that.
Birne Binegar: Okay, so we will have our.
Birne Binegar: First exam a week from today you'll be in class.
Birne Binegar: It will be in class and.
Birne Binegar: What i'll do today, although I said it was a quick review it may very well take half the lecture because I want to touch on all the points that we've covered since the beginning of the Semester.
Birne Binegar: that's a good thing to do, the weekend before the exam until all sort of run through what we've covered so far and then what i'm going to do today and Monday is i'm going to talk about the next topic in the course which is sub spaces.
Birne Binegar: So.
Birne Binegar: we'll do that now, I will probably also talk about sub spaces, on Monday, but Monday, will also be a good opportunity, hopefully.
Birne Binegar: homework set four and five are due on Monday, if you have any issues with those problems such you should.
Birne Binegar: raise them well, you can raise them now, if you have a have any questions, but not certainly by Monday would be best time to get a homework problem straight down any difficulties there.
Birne Binegar: Then Wednesday of next week we'll do a full review i'll walk you through a sample exam and show you.
Birne Binegar: yeah effectively that the exams, even though they can be long, particularly when there's a lot of real reduction that's sort of the tedious calculation.
Birne Binegar: i'll try to show you like explicitly working through exam sort of the most efficient way to get through one of the exams, and in fact what you'll find yeah if you follow the practice exam and know the material that my exams were.
Birne Binegar: Just for doing the calculations alone, you should be able to do that in a half hour, and so you have next or 20 minutes to actually think about things a little bit, but.
Birne Binegar: What you'll what i'm what I typically do, though, is, I give you a practice exam and the exam I give is based on the practice exam rather than homework or other things, so if you.
Birne Binegar: understand how to do the practice exam those questions about what to do on a given problem they should have already been resolved for you and you just have to carry it out.
Birne Binegar: Okay, so we'll talk more about the.
Birne Binegar: The format of the exam next Wednesday, let me.
Birne Binegar: First of all, pause and make sure any outstanding questions.
Birne Binegar: Okay, so.
Birne Binegar: What I want to talk about first then it's the things we've covered so far, we started off the course with you enter the course with a variety of.
Birne Binegar: Definition definitions of what a vector is the quantity with magnitude and direction it's a directed Lang segment.
Birne Binegar: etc, but what we specialize to immediately in the beginning of the course is thinking of vectors as ordered list of numbers, because that first of all, very concrete and seminal it's the essence of all the calculations, we do in the course.
Birne Binegar: So we introduced actors and how to manipulate them.
Birne Binegar: We had vector addition, of course, you known how to add vectors ever since you took calculus one I would affect However, I want you to also think of them a little bit more abstractly as well.
Birne Binegar: Think about what the operation, does it.
Birne Binegar: takes pairs of vectors and it produces a new vector and similarly scale or multiplication is an operation it's easy to perform but also think about it exactly you take a number and a vector and you get a vector out of it.
Birne Binegar: And then we had linear combinations of vectors which basically means combining these two operations to create more vectors.
Birne Binegar: So typical your steel are multiplying vectors and you're adding them together to produce new vectors In fact this is the most general construction that will that we use in this course, and we use it over and over and over again, to create linear combinations.
Birne Binegar: We also have the Doc product which.
Birne Binegar: is useful for a couple reasons, first of all, it gives us geometric information about these ordered list of numbers.
Birne Binegar: Using a dot product actors, we can detect not only the length of the vector whether it's a 10 dimensional vector or a two dimensional vector and we can also calculate the angle between two vectors using the Doc product.
Birne Binegar: But you to think of it as a as an operation that takes a pair of vectors and just takes out a number.
Birne Binegar: Alright, so that was the basic apparatus that we're dealing with now, with the exception of matrices that come in just a second, the first thing I did with this material is i've.
Birne Binegar: talked a bit about the geometry of vector spaces, but it's it's kind of extravagant they call it.
Birne Binegar: geometry because it's so simple, the object the geometric objects, we consider in this course, are not.
Birne Binegar: Things like tetrahedral spears or tauruses or other fancy geometric objects, rather than either points lined planes or generalizations of those objects they're just hyper planes.
Birne Binegar: And they're all constructed, the same way, you get hyper plane is going to be a set of vectors where you start off at a point and then.
Birne Binegar: You take a linear combination of a given list of vectors and you let those coefficients very over the entire real line and then.
Birne Binegar: The set of vectors there's going to be an infinite set of back just because you have an infinite number of choices for each of these parameters, do you want to do Okay, you get a.
Birne Binegar: Large set of vectors and in fact it's either going to be a point that's if you don't have these terms you get a lie, if you have just one vector from which you can depart from the zero.
Birne Binegar: and, more generally, you have higher dimensional hyperplasia but all based on this simple construction that itself it's built from vector addition and scale or multiplication.
Birne Binegar: OK, and then.
Birne Binegar: One of the first utility of this simple geometry what's the fact that if you have a system of equations or that you just have a single equation.
Birne Binegar: In and variables that you can associate with this equation, a certain hyper plane.
Birne Binegar: And in fact what it is it's a hyper plane in the space of variables, you can use this equation to say solve for X m in terms of the other variables.
Birne Binegar: And so, so X one X am i'm thinking here is three variables in the solution to this equation, and what you see is that.
Birne Binegar: As soon as I solve for X M and use that to replace the image component you're able to expand an arbitrary vector that satisfies the equation.
Birne Binegar: In terms of a constant vector and some parameters times other constant actors that's exactly of this form.
Birne Binegar: So solution sets of a single linear equation is always a hyper plane.
Birne Binegar: Of dimension one less and the space of variables here count your variables, you have three variables, you have N minus one of them.
Birne Binegar: But each of the vectors is going to be m dimensional vector until effectively, you have a N minus one dimensional solution set embedded inside of n dimensional space of variables for this equation.
Birne Binegar: Alright, and then, when you have more equations geometrically that just corresponds to intersect the hyper planes, but the intersections of hyper planes are also hyper planes they'll they'll still be sets of that form, even when they're intersections of larger hyper planes.
Birne Binegar: Okay, and so.
Birne Binegar: The next gadget we introduced in the course were matrices I introduced them by first.
Birne Binegar: Talking a little bit about linear systems as on the last slide and then.
Birne Binegar: Where we solve these linear systems, our technique was essentially to manipulate the equations until we got the equations of the solution.
Birne Binegar: However, that was not very efficient, so we developed an ultra another method that did exactly the same thing we would take the equations.
Birne Binegar: We would take the data, the coefficients and the numbers on the right hand side and move your range those into a matrix and then we can manipulate these equations by instead working with the simpler object just.
Birne Binegar: Just a matrix and the idea.
Birne Binegar: Well, well, this is how matrices came into the game, I guess, I should explain a little bit more about matrices before I describe what we do with augmented matrix ease so at this stage of the course, I just wanted to introduce this notion of the matrix so.
Birne Binegar: it's basically a rectangle arrangement of numbers, with certain fixed number of rows and a fixed number of calls and we learn to how we can multiply these guys together.
Birne Binegar: A matrix product, a times B, first of all, you have to remember that it's only defined when the number of columns of the first doctor equals the number of rows of the second factor.
Birne Binegar: that's because the individual entries of the product are determined by.
Birne Binegar: God product and to take a product to two vectors they have to have the same dimension, so that restriction that the number of columns of a is equal to number of rows of be that's just so that these products are defined.
Birne Binegar: But if that's true the Doc product will be defined and you'll be able to count calculate all the entries product of two matrices.
Birne Binegar: And then.
Birne Binegar: Besides.
Birne Binegar: Being able to multiply make these together, we can also add individual matrices together we can scale or multiply them as well, but that works, essentially the same way as it did for vectors you when you add.
Birne Binegar: vectors they have to have the same dimension, and when you add matrices they have to have be the same number of rows and the same number of calls, but the operation is the same you just add similar components of both factors to get the.
Birne Binegar: summit two matrices and you scale multiple you multiply every entry of a matrix by a particular number to carry out scale or multiplication of matrices.
Birne Binegar: And then last basic matrix operation with the transpose and that's simply where you take a matrix and you convert it to rows that were written horizontally you you write down another matrix where the road of the first matrix have been converted into columns.
Birne Binegar: Alright, so now, I get the solving linear systems, so the idea was that there were things we can do to equations that don't change the solution set.
Birne Binegar: On so on a formal level the way we're trying to solve linear equations it's not by focusing on variables that rather focusing on the entire set of equations until we get the equations on the solution.
Birne Binegar: So there were things we could do two equations like interchanging the order equations will play equations by number replacing an equation by at some with another equation.
Birne Binegar: Those elementary operations wouldn't change the solution set but we found that was a much simpler way that implementing them that could be very rapid and that was by instead of working directly on the equations.
Birne Binegar: Going back and working instead on the equivalent dig these augmented matrix ease.
Birne Binegar: And in that guys the things we could do two equations became operations, we can perform directly on the augmented matrix seas.
Birne Binegar: They were called real operations, because each each row of the augmented matrix corresponds to an equation.
Birne Binegar: So when you do something on the equation side there's going to be an equivalent operation that you can do on the augmented matrix side and that's what we focused on so we had.
Birne Binegar: three different things we could do to an augmented matrix that would replicate actions on a set of equations that did not change the solution.
Birne Binegar: All right, so my.
Birne Binegar: plan is to systematically convert a given linear system into the equations of the solution, but the way we do it is we work with the augmented matrix rather than the equations themselves.
Birne Binegar: We use three operations to systematically transform the augment the augmented matrix you start off with good the augmented matrix of the solution and there's.
Birne Binegar: Maybe three basic steps to that first of all, you need to reach relational on forums, so that simply means that the pivots pivots are.
Birne Binegar: Really, a crucial concept.
Birne Binegar: How each athlete identify the special forms of the augmented matrix of the pivots are always see.
Birne Binegar: First Non zero entries in a row, and what we try to do if we try and solve a linear system is our first task is to get the.
Birne Binegar: augmented matrix in re enroll Echelon form so.
Birne Binegar: When you started off with a matrix you might not have zeros below this pivot in the first call, but then you can use operations like this, they clear out the entries below and you systematically work your way from the.
Birne Binegar: upper left of the matrix down to the lower right clearing out entries below pivots and also moving the pit off to the right as you descend through the matrix.
Birne Binegar: Alright, so that's the first stage and, in fact, once you reach this stage the puppets don't move any further well there's more row reduction to do before we reach reduce cholesterol on form, which is something like this, but.
Birne Binegar: The positions of the pivots don't change to get it this matrix in this form, what you have to do is use row operations like.
Birne Binegar: scale or multiplication to change whatever numbers there to make it one, so if there were three I would divide this row by three so that I can get a one or its pivot and you do that, for all your rows and then, in addition, you try to clear out.
Birne Binegar: This.
Birne Binegar: Passing the relational on form what you're doing here is you're clearing out entries below the pivots.
Birne Binegar: In passing, to reduce yo excellent form, besides making your pivots equal to one you also want to clear out the entries above pivots and that's again going to be using this third elementary row operations, for the most part.
Birne Binegar: All right, so.
Birne Binegar: The idea for solving linear system is to convert well My guess is on the next slide you convert your system of equations you represent it as an augmented matrix, then you wrote reduce the relational on form.
Birne Binegar: And then your strategy changes in this part of the reduction you work from the upper left lower right.
Birne Binegar: Then, to get to reduce your rational inform you start in that bottom right corner and you clear out entries in the low above above pivots in the lower rows and you work your way up to the upper left.
Birne Binegar: But once you've done that you reach a unique matrix and reduce row Echelon born and that's going to be the augmented matrix that corresponds to the solution of your linear system.
Birne Binegar: So once you've read this the next day to athlete or write down the solution, so you convert that to equations and in doing so.
Birne Binegar: You make a distinction between two kinds of variables fixed variables, these are going to correspond to columns of The reduce cholesterol on form, but in fact already.
Birne Binegar: identifiable when you reach relational on form because.
Birne Binegar: let's go back to your previous slide once you reach real isilon form the position of the pivots is not changing until the columns they live in are not changing any further either.
Birne Binegar: So already at this stage, you can count the number of free variables in your solution you just look for the columns about pivots when you were in real Echelon doesn't give you the solution, but it tells you what the dimension of the solution.
Birne Binegar: To get the solution itself, you have to proceed further to reduce electronic form, youth, the position of the of the pits do identify which variable she should regard as fixed and which variables, you should regard is free.
Birne Binegar: Then.
Birne Binegar: At this stage, you you take the augmented matrix in reduce your Echelon form and you go ahead and write down the corresponding equations.
Birne Binegar: But what you use those equations for us to express the fixed variable in terms of free variables in any given a row of that augmented matrix and we do shuttle and form it only has one pivot.
Birne Binegar: And the other, the other wrench other non zero entries, we have are going to be attached to three variables, so we just simply move the three variables, to the other side of the equation, and then we have the fixed variables expressed in terms of the free variables.
Birne Binegar: So.
Birne Binegar: Once we convert back to equations we move the free variable say over to the right hand side of the equation, and then.
Birne Binegar: you write down a typical solution vector substituting where you can for the fixed variables.
Birne Binegar: The ollie express now in terms of free variables and you end up with a solution factor that now depends only on the free parameters last step of this is to expand the solution vector in terms of the free parameters and then you'll get that hyper plane that corresponds solution yeah.
Birne Binegar: yeah yeah yeah so let's let's start with.
Birne Binegar: A matrix that said.
Birne Binegar: that's right detriment 01021001.
Birne Binegar: minus 1200000 net okay so it's only good um yeah.
Birne Binegar: Well, we can put something non trivial no I can't I used up all the tickets okay so.
Birne Binegar: Just do this one okay so.
Birne Binegar: Here we have the pivot in here we have a pivot so that corresponds to X two X three So those are going to be fixed variables.
Birne Binegar: right because they correspond to columns that have pits.
Birne Binegar: That means that leaves X one X four and three variables, because in this augmented matrix in Russia long form, there are no pits in these the first and last column of the coefficient side of the augmented matrix.
Birne Binegar: Right so let's write down the equations X two plus two times X for has to eat one second row says that X three minus X four has to equal to, and the last row says zero is equal to zero so it's.
Birne Binegar: kind of a trivial equation it doesn't put any restrictions that all of my variables, but zeros occur during the production process if you have equations that have a relation not amongst themselves.
Birne Binegar: So.
Birne Binegar: i've got I guess i've got me.
Birne Binegar: yeah i've got the equations and i'm like no let's express the fixed variables in terms of the free variables so fixed variables X two X three x two is going to equal one minus two X four X three from the second equation.
Birne Binegar: is equal to two plus export remember the flip side when you move the variables, to the opposite side.
Birne Binegar: And then you write down as typical solution vector.
Birne Binegar: What you'll do is for the fixed variables you'll substitute the right hand side of these two equations but for the.
Birne Binegar: The free variables you don't make any substitution so X one like first column has no pivot so it's free I don't change it X two can be expressed in terms of X four, so I write it as one minus two X four X Ray can be expressed in terms of X for two plus x four and.
Birne Binegar: Export itself is free.
Birne Binegar: Alright, so there's.
Birne Binegar: The functional form of any solution better it depends on the free parameters X one X, for the last step is to expand that solution vector.
Birne Binegar: In terms of the free variable so we have terms that don't involve three variables nothing in the first lie one in the second slot to in the third slot zero and, of course.
Birne Binegar: Then we have.
Birne Binegar: X one terms and in fact that only occurs in this first slot so at 1000 and the X four.
Birne Binegar: it's not appear in the first lot with coefficient minus two in the second slot coefficient one, and the third coefficient one.
Birne Binegar: X four Okay, and so there are you now, we have a solution vector expressed as a point on a hyper plane.
Birne Binegar: here's your starting point here at the two directions, from which you can leave the starting point and this free parameters just a number that could not you could bury over the entire real line to generate a two dimensional plane of solution.
Birne Binegar: Okay i'm sorry if you can't you can't see get from there.
Birne Binegar: But um okay so.
Birne Binegar: Okay, other questions.
Birne Binegar: Oh yeah there's one caveat to this procedure, sometimes you have equations that don't have any solution the way you identify those situations is when you have an augmented matrix where you have a non zero entry, the last call and zeros everywhere else.
Birne Binegar: explain that caveat.
Birne Binegar: So if you have a matrix like this 100000100000 let's put some numbers to one one.
Birne Binegar: I can look at this matrix and I know immediately there's no solution.
Birne Binegar: How am I, seeing that i'm looking at this last row, and if I write down the equation that correspond to this last road zero times X one plus zero times X two.
Birne Binegar: Plus zero times X three is equal to one well simplify the right hand side or left hand side you'll get zero is equal to one and that's a contradiction.
Birne Binegar: The mathematical contradiction.
Birne Binegar: Until.
Birne Binegar: This augmented matrix implies a equation that has no solution and still since this augmented matrix.
Birne Binegar: Has the same solution as the original system of equations the fact that you can't get fed find any solution to the last equation meanings that your original set of equation had no solution either so.
Birne Binegar: This is how you proceed if you have a solution if you don't have a solution, what will happen is, as you reduce the matrix you end up with it could be any number here could be 105.
Birne Binegar: anytime you have a nonzero number in this last column and zeros everywhere else you can end up with an equation that says zero is equal to a nonzero number and you'll have a contradiction.
Birne Binegar: So yeah so this caveat at the end is just so.
Birne Binegar: If you have a contradiction in your equations, how do you recognize it as soon as you spot a row like this, even if you only have started row reduction, you could stop because further row reduction is not going to get rid of this problematic equation okay.
Birne Binegar: Alright, so.
Birne Binegar: that's how we solve linear systems.
Birne Binegar: And then the last topic we covered were in verses of square matrices.
Birne Binegar: Remember it's just a matrix that when you.
Birne Binegar: A matrix has an inverse if you can find another matrix which, when you multiply the two matrices together, you get the identity matrix.
Birne Binegar: And way we develop that sorry going right over the crap blackboard.
Birne Binegar: The way we.
Birne Binegar: found a way of calculating the inverse of the matrix is we use this auxiliary idea, the notion of elementary matrix season what that did, is it allowed us to.
Birne Binegar: Implement elementary row operations using matrix multiplication.
Birne Binegar: That was important to us because, if we can reduce a matrix to the identity matrix then.
Birne Binegar: We would have a sequence of matrices that we can multiply the original matrix and produce the identity matrix so that sequence of matrices.
Birne Binegar: That product is the inverse so that's how we discovered how to calculate the inverse is how to figure out what sequence a row operations would take a to the identity matrix and then by forming the.
Birne Binegar: Corresponding matrix seeds, we can multiply a buy a product of major cities and produce the identity matrix and then, finally, we develop a way that we didn't have to identify the individual.
Birne Binegar: Elementary matrices in this process, all we had to do is take this matrix rule reduce it to reduce your national on form and whatever occurred in this last factor during that process that would be the inverse of a.
Birne Binegar: bit more precisely.
Birne Binegar: Through sounds like the inverse of the matrix you would join to it, the identity matrix of the same size and then you wrote reduce.
Birne Binegar: This double matrix to which we do celestial on form now when you reach the redo shuttle astronaut forum you look at it if, on the right hand side a has been transformed into the identity matrix, then you know that the left hand side of that, let me.
Birne Binegar: introduce some new location here so like not pointing to different matrices so let's call this we do show Echelon form of this matrix let's call it a double prime a join with I double problem or a double prime is whatever happened well and that's what has to happen.
Birne Binegar: Is is that if the reduce cholesterol and form of this matrix has the identity matrix on the left hand side, then this guy.
Birne Binegar: is going to be a the minus first, on the other hand.
Birne Binegar: If this whatever happens to the right hand side is not equal to the identity matrix, then that means there's no inverse.
Birne Binegar: So.
Birne Binegar: The calculation of a matrix inverse.
Birne Binegar: is straightforward all you have to do is take a given matrix and reduce it to reduce electronic form, and then grabbed the right hand factor.
Birne Binegar: The thing you have to worry about, though, is that this guy doesn't necessarily always reduce to the identity matrix and if it doesn't.
Birne Binegar: This thing the reduce cholesterol on form of this matrix does not have the identity matrix on the left hand side that simply means that the matrix does not have an inverse.
Birne Binegar: OK, so now it is straightforward calculation once you know how to carry out reduction.
Birne Binegar: And you get a decisive answer as well.
Birne Binegar: Okay.
Birne Binegar: well.
Birne Binegar: I think that Oh, we have.
Birne Binegar: Well, fortunately, this is maybe not showing up completely on screen, maybe it will Okay, the last thing last topic to topic of monday's lecture was this fundamental theorem matrix in versus.
Birne Binegar: it's fundamental in the same sense that the fundamental theorem of calculus is.
Birne Binegar: really important to calculus it tells you that any derivatives are the opposite of derivatives or integrals or the inverse process of picking derivatives it connects to seemingly very different things.
Birne Binegar: The fundamental theorem of algebra it's a statement that you can solve algebraic equation by factoring the polynomial on one side of it, and so you really like polynomials to solution sets up.
Birne Binegar: equation to holly factor is related to their solution set here we have the fundamental theorem and linear algebra yeah call fundamental pyramid.
Birne Binegar: matrix inverse isn't it relates a bunch of seemingly diverse ideas, the existence of an inverse the ability to find a unique solution to a corresponding linear system.
Birne Binegar: yeah.
Birne Binegar: The fact that.
Birne Binegar: If you have a particular linear system where you have the zero vector on the right that you're only allowed one solution if.
Birne Binegar: The matrix has an inverse also the know that the reduced Ashkelon form of a matrix is has to be equal to the identity matrix.
Birne Binegar: That sort of clear from the calculation I just erased how we calculate the inverse of the matrix which we reduce a joint with I to relax a long form and if a reduced I would ever happen to the right hand factor was the enders.
Birne Binegar: Then, then, finally I guess.
Birne Binegar: A can be factor is is a product of elementary matrices is another.
Birne Binegar: consequence of a having an inverse but should we have a series of five statements that.
Birne Binegar: Maybe these two are closely related, but the other, the other four statements.
Birne Binegar: are pretty disbarred yeah and what they're saying yet if any one of these is true, the other is true So yes, I.
Birne Binegar: know that I only have one solution of such an equation, then I already know that the matrix has numbers I don't have to calculate it well, unless I needed explicitly.
Birne Binegar: This kind of thinking now where we begin to relate calculations that we've done.
Birne Binegar: In separate context, this is going to be where we go next in this course will start tying things together, a lot more closely, yet.
Birne Binegar: What we able to do is take consequence of one statement and infer from that another property of the matrix.
Birne Binegar: Okay, so.
Birne Binegar: Good so that's that's my real quick review that summarizes what we covered since the start of the Semester and that will be the material that will be on the exam.
Birne Binegar: Oh, let me pause here any questions.
Birne Binegar: Okay, so i'm going to make a little bit of headway on substations day I have about 15 minutes left, so I won't get too far on this, but we'll we'll make more progress on Monday.
Birne Binegar: Let me just remind you that.
Birne Binegar: We have a set of vectors in our end linear combinations are just new vectors you can create like combining the operations of scale or multiplication and vector addition.
Birne Binegar: Alright, so.
Birne Binegar: I talked about hyper planes since the beginning of the course i'm going to focus now on a special case of hyper planes so we'll save that as.
Birne Binegar: Well, actually i'll introduce it differently, a span of a set of vectors it's going to be the set of every vector you can reach in the entire space, by taking linear combinations of these guys now if you only have one direction you only have one direction.
Birne Binegar: Then, the only place, you can go from this guy by taking scalable applications of it and adding it to itself or points along the same line until.
Birne Binegar: I do have a single vector span of a single vector is just going to be alive, but um it actually has.
Birne Binegar: Not quite a hyperlink so this should be the form, so what i've written down here is all like during these coefficients.
Birne Binegar: Effectively, constructing a set that contains all possible linear combination, so the given this vectors.
Birne Binegar: it's not quite a hyper plane because it's missing this initial term, and in fact that's the crucial point that makes.
Birne Binegar: A span of a set of vectors, it is an example of a hyper plane only have to do is choose P zero to equal to zero vector and then this construction just replicate this, but a P zero is not equal to zero, then effectively it's sort of offsets.
Birne Binegar: This hyper planes gets sort of displaced from the origin, so if you had a hyper plane that passes through the origin which notice this set by choosing a number see one two ck equal to zero.
Birne Binegar: You will have the linear combination for every doctor is multiplied by zero, which is going to be the zero vector so this span always contains the zero vector but more general hyperplasia doesn't necessarily contain the.
Birne Binegar: The zero vector.
Birne Binegar: here's a way to think about it, let me just work with lines.
Birne Binegar: here's a general line in the plane.
Birne Binegar: But it's it's not going to be a span of vectors because it doesn't contain the origin.
Birne Binegar: But you can draw a bunch of parallel lines, none of which actually contain the origin until you reach the line that passes through the origin.
Birne Binegar: This guy is going to be a span of the vectors, but these other vectors that essentially are flying heading in the same direction they're just displaced a little bit from the origin, those are not going to be spans of vectors.
Birne Binegar: What we'll see here is that.
Birne Binegar: My comment from early on, when I said that these hyper planes a little vector spaces sitting inside larger back your spaces that's not quite right.
Birne Binegar: They look like little vector spaces sitting inside the larger vector spaces, but they failed to be vector spaces, because you cannot perform operations of scale and multiplication and vector addition on them consistently.
Birne Binegar: And so, this whole point of the next topic which is sub spaces it's the specialized to hyper planes without this.
Birne Binegar: offset here and we get rid of that guy then attack these hyper planes are going to act exactly like.
Birne Binegar: vector spaces, we can scale a multiply vectors and stay inside the same hyper plane, we will be able to.
Birne Binegar: Add vectors inside the hyper plane and they'll still live inside the hyper plane, so what these guys the span of a set of vectors what it actually represents is the set of vectors that looks like.
Birne Binegar: Another vector space, but in fact also it meets the action of scale and multiplication and vector addition.
Birne Binegar: they'll that's that's the purpose of the definition of the subspace that i'm about to give but it takes a little development to get to that.
Birne Binegar: The idea in the end it's going to be, we want to look at special subsets of vectors inside our end that look like hyper planes.
Birne Binegar: But in addition they pass through the origin, so you don't have this offset there what that will allow us to do is scale and multiply vectors and stay inside the set or add vectors and will still stay in touch on the set.
Birne Binegar: Alright, so let me make this whole.
Birne Binegar: Hopefully, a little bit clearer i'm making a little bit more formal.
Birne Binegar: So now.
Birne Binegar: let's move this.
Birne Binegar: Here we go all right alright so.
Birne Binegar: First of all, my operation on a set I guess mean some procedure or some function that takes elements of the set and produces something else but.
Birne Binegar: Our focus here is going to be on operations that are closed don't what what I mean by that is you take an element of your set you do something to it, but you don't go to a new space you stay inside the same space you started off with the same set you started off.
Birne Binegar: So.
Birne Binegar: we've seen this before the scale or multiplication it takes a vector inside our end well also uses a number, but the result of multiplying a vector.
Birne Binegar: scalable vector by Lambda is to produce a new vector so.
Birne Binegar: anytime you have a vector and a number, you can create a new vector that lives in the same vector space has the original vector space.
Birne Binegar: So R n is said to be closed under scale multiplication you do a scale or multiplication you don't leave or N and similarly.
Birne Binegar: When you have two vectors that start off in the N dimensional vector space, you can add them, but that result of adding those factors does not take you outside the space.
Birne Binegar: Alright, so let's form like this will say that a subset of ours vector space of n dimensional vectors is closed under scale or multiplication if whenever you scale or multiply a vector inside the set you don't leave the set.
Birne Binegar: And it will say a subset of our end is closed under vector addition if, after you add two vectors that start in the set you're still inside the same set.
Birne Binegar: Now that looks really natural and you might think that well, it must be lots of substances, but in fact those two properties are actually kind of hard to maintain Let me give you.
Birne Binegar: Well yeah here's here's the definition of suspect subspace of our end is a set that's closed under both those operations, the fundamental operations of linear algebra scale multiplication in Dec tradition.
Birne Binegar: You can think of Sub spaces as be little vector space is sitting inside larger vector spaces so subs basis is sort of like subset subset it's a smaller set sitting inside the larger set well substations.
Birne Binegar: More so than hyper planes are really can be thought it's legitimate vector space is sitting inside a larger vector space, because we can perform scale and multiplication and vector addition without leaving no set.
Birne Binegar: Let me give you some examples of things that might think of a nice subsets of sight of the plane that happened not to be sub spaces let's take the circles.
Birne Binegar: All the points on the unit circle or point where they distance from the origin is just one.
Birne Binegar: Right so X squared plus y squared at the dot product this a factor of itself, so it was just saying that link we're looking for vectors of linked one.
Birne Binegar: And it's not closed under stealer multiplication because, if I choose a number two and I multiply a vector that does live inside this set one squared plus zero squared does equal one.
Birne Binegar: So this guy lives inside this set here's a number let's multiply them together well I get a new vector but now it no longer satisfies the equation to live in inside this unit circle, some of the squares with the components here is going to equal four rather than one.
Birne Binegar: Alright, so the circle is not closed under scale or multiplication, nor is it closed under vector edition Here are two points.
Birne Binegar: Of the not vectors of unit length they both live on that circle, when I add them together, I get a new vector but now it's link is going to be the square root to rather than rather than one.
Birne Binegar: Okay, so now you just start writing down random set yeah particularly ones that you think that are interesting geometrically not likely at all that you're actually going to be sub spaces.
Birne Binegar: Let me do another example.
Birne Binegar: The set key that i'm creating by taking pairs of integers and using them to form two dimensional vectors that is halfway towards a sub space, but not quite.
Birne Binegar: It is closed under back tradition, because let's take two vectors in the set accuse an integer an integer that will be my first Dr another energy or another energy that will be a second factor.
Birne Binegar: If we add the two vectors together each component it's going to be a sum of integers and so.
Birne Binegar: result of adding to arbitrary doctors inside this set is a vector whose components are still integers, and so this set T.
Birne Binegar: form by using integers for the entries is going to be closed under vector edition.
Birne Binegar: But it's not closed under scale or multiplication because, if I decide to take a vector inside the set and multiply it by a real number, like the square root to Well now, the new component is no longer an integer and so it's not closed under scale or multiplication.
Birne Binegar: And so, from the point of view is linear algebra this set it's not going to be a substantial, we should try to think of it as a little vector space sitting inside a larger vector space.
Birne Binegar: What about hyper planes hyper planes where our favorite geometric objects now can they be thought of as a sub spaces well i've already pointed out, this this initial term causes a problem well i'll show you how that works out.
Birne Binegar: If you have a have a set of vectors ah that's actually a hyper plane, if you if.
Birne Binegar: This hyper plane does not contain the zero vector then when you scale old client any vector inside the hyper plane by the number zero.
Birne Binegar: You always get the zero vector but if zero is not sitting inside that hyper plane you've left the hyper plane you're in this situation.
Birne Binegar: If if the here's a here's on one dimensional hyper plane on the blackboard if it doesn't contain the origin, then it can't possibly be.
Birne Binegar: subspace because scalable applying any of any vector along this line by zero is going to move you to a different a different subspace effectively or different hyper plane.
Birne Binegar: Okay, so.
Birne Binegar: The examples i've given so far, make it look like it well substations must be really rare, but in fact the useful the usual set in linear algebra on fact almost always sub spaces, or even these hyper planes, you can think of it as a subspace.
Birne Binegar: Then i've offset like you can think the basic instructions to get that subspace and then, if you if it doesn't subspace isn't quite what you want, then you fixed it up, but typically what we're doing in linear algebra constantly is working with.
Birne Binegar: Not so much individual vectors anymore or principal objects that we're going to study henceforth it's going to be sub spaces.
Birne Binegar: And they're going to be effectively things that we can really jump on regard as actual vector space is because we can perform scale or multiplication and vector addition without leaving them.
Birne Binegar: So these sub spaces are always going to be vector spaces literal vector spaces inside larger vector spaces well hyper planes.
Birne Binegar: You have this problem that if it doesn't contain zero well then you're not going to be closed under scale or multiplication.
Birne Binegar: Okay, well, maybe i'm maybe that's good enough for today.
Birne Binegar: What i'm going to do on Mondays i'm going to give you more examples of substations no show you how they arise naturally by linear algebra.
Birne Binegar: So, for example.
Birne Binegar: One way to create a subspace to get rid of those starting point, if you have a linear common.
Birne Binegar: You have a span of a set of vectors set up all possible linear combination it's always going to be a sub space, and so one way of making some spaces, is just to.
Birne Binegar: Consider linear combinations of vectors with no asset, such as you take all the rows of a matrix and you consider all the.
Birne Binegar: Factors you can reach by taking multiple scale linear combination of rows you're going to get a subspace take the columns of a matrix you take all possible in your combinations of columns you generate a state.
Birne Binegar: there's no.
Birne Binegar: there's no cost so that you won't have any offset there you'll again get a subspace so.
Birne Binegar: At first substations are going to look very common, then I show you that they're actually pretty rare, but now what i'll show you on Monday, is that the usual sub spaces are also very common and easy to work with.
Birne Binegar: OK so i'll stop here today and wish you a happy weekend to have a look at the material.
Birne Binegar: And so any questions I can address on material that's going to be covered, on the first example like happy to talk about it link on Monday, we will do a general review for review on Wednesday.