Birne Binegar: Okay welcome back to 3030 hold on.
Birne Binegar: Alright, so um let's see, let me share the screen.
Birne Binegar: Hope y'all can hear me i'm getting his thing y'all set up going okay so um.
Birne Binegar: Today, what i'll do is i'll do a quick recap, of what went on on Monday and Tuesday this week and then we're going to explore.
Birne Binegar: A third operation, we can perform on vectors, we can already add them to multiply but there's another operation called dot product that is useful for extracting geometric information about vectors.
Birne Binegar: So all discuss that and then we'll conclude this week's lecture with a discussion of the really basic geometric optics that occur in linear algebra they're very simple and nothing as exotic as even a sphere occurs naturally linear algebra rather what we.
Birne Binegar: focus on our real points lined and black planes were in generalizations of those objects Okay, so let me begin by asking if there any questions.
Birne Binegar: I will point out, initially, I had the first homework assignment due next Wednesday i've already delayed that till a week from today so Friday so you'll have.
Birne Binegar: All of the holiday weekend work on the homework, as well as some additional time next week and perhaps on Wednesday next week or maybe do some of the Web assign problems explicitly in class So you see how that mechanism works okay well let's get started.
Birne Binegar: What happened on Monday, Tuesday, well, we introduced the principal objects of this course, which are vectors.
Birne Binegar: which could be a number of things as we've seen quantity from magnitude and direction, it could be directed line segment or, and this is the principle, we will regard vectors in this course as ordered list of numbers.
Birne Binegar: All typically right these.
Birne Binegar: guys by using square brackets, do you know, an ordered list putting the elements of that list inside in their particular order.
Birne Binegar: write down a set of numbers and then close it in square brackets that's going to constitute a vector.
Birne Binegar: All right, what's nice about the directors of course because we're using numbers all the time is they'll be eminently capable, the capital objects, the numbers that we can add subtract dividing cetera.
Birne Binegar: Okay now occasionally will depict vectors as directed line segment but that's primarily not for calculation, but rather, so we can visualize and get someone to it, of understanding with what's going on, otherwise we're just staring at numbers and it doesn't really offer that much intuition.
Birne Binegar: Alright, so location I use this funny are you know set of real numbers and if I put a super script there i'm looking into different object i'm looking at the set of all possible n dimensional vectors which you should interpret it is all possible ways of writing a list of in real numbers.
Birne Binegar: Okay, so and then then here's our set theoretical note set theoretical location will use throughout the Semester.
Birne Binegar: The way I construct sets is first of all make sure they're explicitly sex buying closing the elements inside curly braces distinct from the square braces up here and then for this set of all n dimensional vectors well each vector is going to be a list of in real numbers so.
Birne Binegar: here's a list of things.
Birne Binegar: And then, this middle of.
Birne Binegar: This vertical flash in the middle of to be read as such debt so aren't in if the set list of in objects, where each of the objects, the one to be n is real number oh it's a.
Birne Binegar: long way of stating verbally but it's simple way of denoting mathematically through our new location what our principal objects are listed in real numbers.
Birne Binegar: Okay, and then we also discussed the two fundamental vector operations they're very simple, we can add vectors that simply by adding similar components to one another in writing the result as another list of in numbers.
Birne Binegar: And scalable application is very easy as well, you have a number Lambda and N dimensional vector V one to be in.
Birne Binegar: The scale or multiple that vector by the number Lambda just going to be the list of in numbers, you get by multiplying each of the elements of the original list by the number land.
Birne Binegar: Okay now.
Birne Binegar: One nice thing about these two formulas here.
Birne Binegar: Is that once they're established declared.
Birne Binegar: They satisfy a number of easy identities that basically are all boil down to properties of real numbers, so let me start listing them.
Birne Binegar: Now so first of all, when we add vectors the order in which we write the vectors down, whether we write you pledge D or B plus you is immaterial to the result that property is actually a property of this edition is called Community of any a vector addition.
Birne Binegar: A second property well our definitions tell us how to combine two up for breakthrough addition tells us how to combine two vectors.
Birne Binegar: If we want to combine three vectors using our two vector rule we are going to have to decide on whether.
Birne Binegar: To add three doctors do I first and the two vectors and then the third or do I add the last two vectors before adding the first factor to it they'll these could be different, depending on how on this notion of vector edition is defined but it turns out.
Birne Binegar: That with this notion of vector addition it doesn't matter how you group things, the result of adding three vectors together, using the two vector rule is going to be i'm not gonna not gonna have any consequence you'll get the same result and then how you the vectors then, if you.
Birne Binegar: introduce the zero vector which is just the vector which every component is equal to zero, and you add that to an arbitrary that you, you always get back get back to you so that's called the existence of an additive identity, you can add something to you, without changing you.
Birne Binegar: know we also have added in versus if I add to you, your skill and multiply by negative one, I always get the zero vector so you will have a additive inverse there'll be another vector that I can add to you that will produce zero when in fact it's exactly minus one time she.
Birne Binegar: or property that we can distribute scale and multiplication over the spectral edition, right now, this plus sign now where i'm adding vectors is different than.
Birne Binegar: Simply this not the simple distributed law for real numbers because the addition here is additional vectors but now we can distribute now multiplication over vectors and we can distribute the addition of real numbers over scale and multiplication.
Birne Binegar: And we can.
Birne Binegar: Have compatibility with scale and multiplication factor edition so here what I do on the on the left hand side here is I multiply you ID using scale multiplication.
Birne Binegar: Using this definition right and then and then i'm multiply again by sea and the claim is, I get the same result as if I first multiplied C and D together before actually carrying out the scale or multiplication.
Birne Binegar: And lastly, one times the vectors equal to vector.
Birne Binegar: Now, each of these properties is really simple that quite trivial to prove.
Birne Binegar: Using these definitions of what it means to add vectors and scale and multiply vectors Let me give you an example.
Birne Binegar: So if I want to show that addition is complicated, what i'll do is i'll plug in explicitly for you and be using their expression as ordered lists night till now, you plus D is going to be this list, plus this list.
Birne Binegar: And our rule for combining.
Birne Binegar: list of numbers, the vector edition is to add similar components so result of adding you to be is going to be this vector here, but then.
Birne Binegar: Because inside each of these components we're just adding real numbers, I can use the property that edition is complicated inside the real number, so I can reverse.
Birne Binegar: These songs inside but notice that the plus sign here on the left hand side is edition of vectors but one working inside the vector in terms of its individual components, the plus sign here that i'm circling is now edition of real numbers which is complicated.
Birne Binegar: And then I could take this vector and break it apart into two summons and now you've seen I demonstrated that the vector some of you, plus D it's the same as the vectors none of deep slushy and, in fact.
Birne Binegar: Each of these eight properties can be verified by simply substituting in for the symbols boldface you both hd.
Birne Binegar: ordered list, such as this and then calculating both sides and seeing that in fact it's true the identity is true.
Birne Binegar: All right, so um I don't want this is not such a formal group course i'm not going to have your approval eight identities, but I do want to remark that these former identities are going to be really important to us.
Birne Binegar: About midway through the course and we start to think about vector space is other than order list of numbers.
Birne Binegar: What we're going to find is that anytime we have a set where we have to operations of vector addition to kill a multiplication define so here I want you to think.
Birne Binegar: Well i've given you an example, you can think of oscillation to sound waves is acting like vectors and in fact you add sound waves, by combining sounds together the way individual instruments there sounds combined to create an orchestra and scale of multiplication.
Birne Binegar: has to do with maybe increasing the amplitude of an oscillation or decrease in the application, possibly reversing each phase if you're multiplying by a negative number we've seen that we can.
Birne Binegar: Have we have other SIP SIP behave somewhat like doctors and even have vector addition in scale and multiplication.
Birne Binegar: What will find, though, is that if we have a set some abstract said, well, which we do have some notion of vector addition in some notion that skill or multiplication and, in fact.
Birne Binegar: identities, like this.
Birne Binegar: are now defined for these abstract vectors in your abstract notion of additional skill or multiplication then they're going to behave as far as let me Roger goes exactly like the vector space that we're used to at that point in the course up the stakes are n.
Birne Binegar: and, moreover, if we want to calculate in these more general vector spaces, what will be able to do is exploit the type of connection between our end and or.
Birne Binegar: After abstract vector space and that poor calculations back to this readily calculable setting we're all we're doing is manipulating list of numbers.
Birne Binegar: Okay, so that is more of a preview of the more formal part of the course that comes about midway.
Birne Binegar: But i'm.
Birne Binegar: Like, I just wanted to point that out at this stage in the game.
Birne Binegar: will be considering much more interesting things later on, but the first thing we're going to establish in this course is the calculation of setting upon which everything depends.
Birne Binegar: Okay, so.
Birne Binegar: Okay, I see some people are saying that they could not hear the sound and there were suggestion to exit doom and try again our people back in in capable of hearing me.
Birne Binegar: Okay, good all right.
Birne Binegar: All right, thank you.
Birne Binegar: keep those questions coming in old glanced at the chat every every once in a while and, hopefully, if you speak up i'll be able to hear you as well.
Colin Myers: i'm the type of question Would you mind giving me an example of vector edition like with real numbers oh sure that.
Birne Binegar: it's simple to do.
Birne Binegar: So we're gonna represent that through this list right so here's the factor in a list of two numbers you like to add it to a list.
Birne Binegar: of another peer real numbers well all you do is add a component by component, so the first component of the result it's going to be the sum of the first component.
Birne Binegar: You know the individual sons and the second component is going to be two plus two squared few more I guess minus two squared kid.
Birne Binegar: All right, does that show up, you may have to.
Not.
Birne Binegar: Let me stop the share and then you'll see.
Colin Myers: I can see it, I said, you can see it okay good.
Colin Myers: idea.
Birne Binegar: So in fact that's a real good question because.
Birne Binegar: we're going to talk about vectors and oftentimes i'm just going to say the word of prayer the symbol for a vector down but it's really crucial always say to them as ordered list of numbers in this in this.
Birne Binegar: stage of the course because these are the things we can directly calculate with will get nice concrete answers, so it will really fun provide us with a firm foundation for all the abstraction that comes later.
Birne Binegar: Okay, so.
Birne Binegar: there's one more operation that we can perform what other operations, besides the dot product, but one more principal operation that will.
Birne Binegar: Sometimes have recourse to in this course it's the so called dot productive vectors it's another operation so you're you have to.
Birne Binegar: recall her location, when I put this time symbol this X between to set it means i'm considering pairs of elements one element from the first set paired with another element, the second set.
Birne Binegar: So what God product, the way to think about it it's a certain function that takes pairs of vectors and produces a real number so there's no super script here so taking two factors and it's producing a real number.
Birne Binegar: Alright, so what is it well.
Birne Binegar: You know in this course, well, we think of vectors as ordered list of numbers.
Birne Binegar: will define it in this fashion, so this ordered list dotted with this sort of list produces a number.
Birne Binegar: that number is just the product of the first components of each factor some with the product, the second components some with the product of the third component until we've exhausted all the components.
Birne Binegar: Alright, so notice in this definition of the Doc product as soon as we have a list of numbers, or to list of numbers, one for each factor here we can compute explicitly a numerical value for the Doc product.
Birne Binegar: This product is going to be used to extract geometric information about our order list of numbers.
Birne Binegar: How so well it's because, let me just remark here that i'm.
Birne Binegar: More or less introducing the location, when I slam a bunch of numbers up up up and use this summation symbol to indicate the sun.
Birne Binegar: So what this symbol means is well i'm going to add up a bunch of terms and the way I create that is i'm going to iterate over this index I so each of my vectors as components that are labeled by indexes 123 of the end until what I do to calculate this Doc product is I.
Birne Binegar: multiply together components, with the same index, and then I summon them all.
Birne Binegar: So now that's an ocean now.
Birne Binegar: Rita accumulation of that summation location, but now let me give you the geometric version of what the product is this is the one you probably most familiar with from the physics or engineering class.
Birne Binegar: The Doc product that two vectors it's going to be the link that one vector time the link to the other vector times the cosine of the angle between the two doctors, so you can think of this geometrically here's two vectors and vector a.
Birne Binegar: vector be.
Birne Binegar: And then.
Birne Binegar: The angle between the two vectors.
Birne Binegar: Right and then the statement.
Birne Binegar: In the shoe metric version is I can calculate the Doc product of these two geometric doctors by measuring the length of a.
Birne Binegar: measuring the link to be and then measuring the angle between the two vectors with that I have all the data I need to compute explicitly on my calculator the value of a dumb be.
Birne Binegar: But if I think of a vector as an order list of numbers, I have a different procedure for calculating the same product it's just that I.
Birne Binegar: Some the pairwise product of all the components all right what's nice about this is having two different versions of the product, it allows us to interpret the linear algebraic version geometrically.
Birne Binegar: Joe let me show you um that's going to be really useful to us because Well, I can draw two dimensional vectors as vectors in a plane right if I wanted to draw this vector two three so that's it written so for the live number if I want to draw this.
Birne Binegar: I simply go two steps in the.
Birne Binegar: Horizontal direction three steps in the vertical direction and I draw a direct line segment going from the origin is a point.
Anthony Baumbach: Could you move your paper down, please.
Birne Binegar: There you go this numerical vector can be written it is easily converted into a on.
Birne Binegar: A geometric factor, and when you do so, these two definitions are actually not compatible with one another.
Birne Binegar: But that I can only do this this explicitly when i'm working on a piece of paper.
Birne Binegar: Because it is difficult to draw three dimensional vectors on a piece of paper, and I think impossible to draw a four dimensional thing or on a flat two dimensional piece of paper.
Birne Binegar: And so that I won't be able to measure links and angles, if i'm dealing with four dimensional vectors I have no four dimensional protractor nor a four dimensional ruler to work with.
Birne Binegar: Okay, but because these are all essentially.
Birne Binegar: Both vectors and we can pass back and forth between your description as directed line segment or ordered list of numbers it'll this formula here allows us to ascribe a geometric interpretation to doctors in a higher dimensional space.
Birne Binegar: So here's how we do that we start with a geometric rule.
Birne Binegar: And then we use it to solve for these various German geometric information, for example, if I want the length of a let's take this formula and substitute for the vector be.
Birne Binegar: The vector a and so a dog with a is.
um.
Yaqoub Alkandari: Excuse me, Sir, we cannot see the bottom of the page.
Birne Binegar: Oh, you got your yep.
Yaqoub Alkandari: Thank you sure.
Birne Binegar: So.
Birne Binegar: I can use the geometric formula here, and I can use it, so what i've done here is like, let me start with a new page here.
Birne Binegar: Here, if.
Birne Binegar: A God hey.
Birne Binegar: Sorry that's gonna show up if that's equal to the magnitude of a times the magnitude of a.
Birne Binegar: Time the cosine of the angle, between a in itself.
Birne Binegar: Well, a this angle is going to be zero degrees in the cosine of zero degrees it's going to be factor one, and this is just the length.
Birne Binegar: square so a dot with a is just going to be.
Birne Binegar: The link Square, but I have an alternative way of computing the Doc product to pay I don't have to actually measure links and angles.
Birne Binegar: I can use the components instead and so now, I have a way of defining the link of an n dimensional vector by simply taking its dot product group itself that's given by this formula and then taking the square root.
Birne Binegar: And Similarly, I can calculate angles, in a higher dimensional states what I do is, I take the geometric formula I solve for the angle well you know i've done some algebra.
Birne Binegar: To isolate data on one side of the equation, but it's coming directly from this definition of the geometric version of the product and then because I have.
Birne Binegar: A formula for the algebraic version of the product just substitute in what you see is that now, if I know the components of the vector a&b as order list of numbers, I can now compute the angle between A and B.
Birne Binegar: So the Doc product.
Birne Binegar: allows us to extend or geometric intuition involving links and angles of vectors we allows us to extend that sort of thinking into higher dimensional spaces, we can.
Birne Binegar: Count we can explicitly calculate the linkedin angles between vectors by evaluating their products we just don't use program you should rulers to measure these things rather we use the components of the vectors to calculate them.
Birne Binegar: Alright, so here's a quick example here's the vector it's a four dimensional vector I do not have a four dimensional ruler, so how i'm going to measure wrestling well, I simply tickets dot product with itself realize that it's dot product itself is the square this link and so.
Birne Binegar: I carry out the algebra solid for them, the link itself, and I see it's equal to the square six because the Doc product of a with itself is equal to six.
Birne Binegar: and, similarly, if I have to four dimensional vectors and I want to know the angle between them.
Birne Binegar: well.
Birne Binegar: i'm going to use the geometric formula, but let's compute the individual pictures here I computed the length of a using the inner product of a web itself.
Birne Binegar: As a thing of as this list of numbers and, similarly, I can calculate the link the beam and then I also can calculate the Doc product A, B and now.
Birne Binegar: Here in the Center here, I have my geometric form a dot B has to be the same as the length of a and the link to be times the cosine of the angle.
Birne Binegar: Alright, so.
Birne Binegar: What i'm doing on the right here is i'm substituting in for the length of a as computed from the components of a.
Birne Binegar: The link to be as computed from the components of be and i'm also substituting on the right hand side, the value of a.be using the components of pain be.
Birne Binegar: Well, so.
Birne Binegar: Since one is equal to a dot B and a tip is also equal to two times a squared is seven times the cosine of the angle, I can use this last equation to solve for that angle.
Birne Binegar: And so, when we do so, we find that the angle between these two numerical vectors is effectively 79.1 degrees, so we can actually do geometry.
Birne Binegar: length and angles, even in a higher dimensional states but we'll use the Doc product to convert length now convert it links and angles, the geometric information.
Anthony Baumbach: Is there a formula sheet with all these formulas on it.
Birne Binegar: Well Ah yes, the lecture slides are all posted on on canvas so yeah you have a complete set of notes for each of these lectures.
Okay.
Birne Binegar: OK, so now.
Birne Binegar: So now, I want to.
Birne Binegar: Think about the geometry of vectors in a different way, oh now show you.
Birne Binegar: How to go from vectors to simple geometric objects and this kind of geometric object we're going to consider here are going to be really, really simple kind of.
Birne Binegar: The basic things about which you make all of geometry we'll start talking about point.
Birne Binegar: Then we may be thinking about it infinite set of point but the simplest construction of an invalid set of points might be aligned and so i'll tell you how you think about lines indicators and they'll generalize that the planes and higher dimensional objects.
Birne Binegar: Okay, so let's start.
Birne Binegar: Okay, so the first point, sorry for the pun, there is that vectors correspond to points in the plane right by point, you know playing, we just need a single dot somewhere existing in the plane and, naturally, these correspond the vectors because, once we choose coordinate system.
Birne Binegar: We draw a direct line segment from the origin, to the point in question we've got that point prescribed by this vector and different point would be correspond with different vector based so.
Birne Binegar: dry things on piece of paper point correspond the vectors and we'll think of this more generally.
Birne Binegar: For example, points in a three dimensional space well to describe those you'd have to give their horizontal.
Birne Binegar: position or horizontal white position and the vertical Z position, then you describe the point in three dimensional space, but that data also constitute an ordered list, but we always use.
Birne Binegar: The first component represent the X coordinate the second component for the like important third component for the Z component will always associate you're given point in space in order list of three numbers and and and but that easily generalized says two.
Birne Binegar: vectors in an n dimensional space will just use and coordinate those in coordinates will then form an ordered list of and numbers and so a vector in our linear algebraic set points and more and more points in an n dimensional space are going to be.
Birne Binegar: Equal then to order lists of in numbers are simple enough now, what about lines lines are.
Birne Binegar: actually consists of an infinite set of vectors.
Birne Binegar: Let me show you.
Birne Binegar: A way of constructing a lot.
Birne Binegar: Usually linear algebra.
Birne Binegar: Right so let's give it a self the coordinate system to work with here's a point now let's look at a line that passes through that.
Birne Binegar: Straight in here.
Birne Binegar: Yes, do you know I mean right wine passes through that point so actually now this line when I convert the vectors.
Birne Binegar: actually consists of a whole bunch of direct line segment right all in reference to this common origin, but you see this line actually contains or corresponds to an infinite set of vectors.
Birne Binegar: Let me show you how you construct each of these vectors using linear algebra.
Birne Binegar: You start with the starting point so let's call this initial point P zero.
Birne Binegar: And then, what i'm going to do is i'm going to add to it be zero a scale or multiple by a parameter T have a certain vector D.
Birne Binegar: So what i'm going to do this i'm going to write D as a vector heading off in this direction so what's the rule for adding vectors geometrically.
Birne Binegar: You take your first factor you write it down and then you take the base of the second factor you put it at the tip of the first factor and you follow that vector out so maybe this point here.
Birne Binegar: This line from the origin to this point might be P zero plus the.
Birne Binegar: But if I make this vector longer by multiplying it by a positive number T.
Birne Binegar: Well, what that means is I stray further from the p zero once I add this vector T times D to it like go further down this way by using larger values of tea.
Birne Binegar: Or if I multiply the by a negative number that's going to not change its overall direction or reverse and we exactly reverses direction so maybe a point here draw the vector.
Birne Binegar: who'd be thought of as P zero Plus minus the right it's gonna head in the same direction as D, but well not quite the same as going to be 180 degrees different than the original vector D and if I make tea a larger negative number, well, it just means i'm going to move further sorry.
Birne Binegar: Further off in this direction, but it'll still be on the same line so we'll think of lines in this course that's a particular set of vectors it's always going to be constructed as a.
Birne Binegar: Starting in a starting point will call P zero and we allow ourselves to stray from that starting point in a particular direction D, but the distance.
Birne Binegar: in that direction D that were strange relieved that arbitrary, so that we fill out the entire line.
Birne Binegar: Alright, so linear algebra a line is a set of vectors of this form of course there's other ways of thinking about that there's.
Birne Binegar: This way we think in this description we're thinking of the line through the point P zero in the direction of D.
Birne Binegar: But there's other ways of describing lines if I give you.
Birne Binegar: exactly two points in the plane there's exactly one line that passes between them right.
Birne Binegar: there's only one way I can put there's only one line is going to contain both of these points.
Birne Binegar: well.
Birne Binegar: If I know these points, I can actually compute the linear algebra eighth line this way.
Birne Binegar: here's what it looks like.
Birne Binegar: it's a it's the set right it's effectively a vector plus an arbitrary parameter becomes another vector itself, the same form if we have here but note that one T is equal to zero.
Birne Binegar: On the second term does not contribute and so when T is equal to zero, the corresponding point is simply the vector a.
Birne Binegar: So every point on the line.
Birne Binegar: As a form a plus T times.
Birne Binegar: b minus a when T is equal to zero what you're going to get is a so this point lives on the line.
Thomas Kidd: We cannot see the paper sorry.
Birne Binegar: And when T is equal to.
one.
Birne Binegar: Well we'll have.
Birne Binegar: A plus one times B.
Birne Binegar: minus.
Yaqoub Alkandari: A then.
Birne Binegar: Here the a's are going to cancel and you're left with be so for special values of tea, we see both.
Birne Binegar: A and point B lie on that line, and so, even when you introduce your line or prescribe a line is passing through two point yeah point on the other, line can all be written in this form.
Birne Binegar: In calculus and other subjects, you might think of a line is a special kind of curve, a way of for each value of T.
Birne Binegar: Getting a point in the two dimensional space and if the components are actually linear functions of tea you actually get line.
Birne Binegar: Oh here's what we might call parameters curve for each value T I get a vector in a three dimensional space and by burying the parameter T.
Birne Binegar: i'm going to get over the entire line i'm going to get an infinite set of factors but it's it's still have this basic form, all you have to do to see that, yes, much take them sure metric for and break it up.
Birne Binegar: If you're online consistency of all the points you could reach by going to see a one plus.
Birne Binegar: He can't do one.
Birne Binegar: Second point.
Birne Binegar: A to was he.
Birne Binegar: He to suggest using linear functions for all the components of this.
Birne Binegar: This parameter ization of the point on the line, what you can do is you can.
Thomas Kidd: We cannot see the paper sorry this is dumb thing.
Birne Binegar: Sorry there that should show it up Okay, so what you can do is collect together.
Birne Binegar: All the components that don't contain T into one term and then add to it all the terms that involve T.
Birne Binegar: She and and then.
Birne Binegar: You can think of this as the vector a and then this is the scale or multiple.
Birne Binegar: Of a vector of components D one through the end by T and so again.
Birne Binegar: Even when you think of a line as a linear curve embedded in a high dimensional space, it still has this basic born a point plus some parameter times a.
Birne Binegar: Another vector.
Birne Binegar: Right so.
Birne Binegar: we're going to think of long will constantly going to rethink of this form as the basic definition of what a line is in linear algebra.
Birne Binegar: Not only because.
Birne Binegar: These other ways of writing down lines are equivalent to this this because this one generalized is very nicely to the higher dimensional objects.
Birne Binegar: So let's consider, for example, what it means to have a plane.
Birne Binegar: Well, to get a plane.
Birne Binegar: or think of this sheet of paper it be a flat two dimensional plane let's pick a starting point and maybe we go a little at one direction on one and now i'm going to allow myself to go, not just a single direction from this point, but two different directions.
Birne Binegar: And when I do so I can reach every point on the entire plane here by going a little bit in this direction and another bit in this direction.
Birne Binegar: For example, the point here how would I reach well, I guess, I would go no idea is to realize this point as be a vertex of a parallelogram.
Birne Binegar: With signs that are parallel to these two guys well I guess it's going to look, let me show you something little bit more this year, so I guess I get a parallelogram on drawing on there that's supposed to be parallel for this wine and beer so.
Birne Binegar: The side here is Parallel to this direction of the two so it's going to be some multiple of the two, and this slide here is parallel to the direction that the one, so it will be some multiple one, and now I got the point.
Birne Binegar: Here in the plane described as a certain multiple that would be one plus a certain multiple back up to.
Birne Binegar: And so that's how we get planes like taking a given point and straying from that in arbitrary disney's on two different directions.
Birne Binegar: Now we can create even higher dimensional objects, all we do is we keep increasing the number of directions, from which we can depart from our starting point, p zero.
Birne Binegar: Right and so now.
Birne Binegar: Now you'll see us that.
Birne Binegar: Whether we're thinking about point lined planes that all going to have the same basic form if the point, a single vector.
Birne Binegar: here's the line it's going to be a single vector plus the ability to add on a vector of arbitrary length on a certain direction that will fall on the line.
Birne Binegar: To get a plane well similar construction, we started with a point, and then we allow ourselves to strengthen that point now in two directions.
Birne Binegar: And for higher dimensional type of playing to we just introduced we're directions direction from which we can depart from our initial starting point.
Birne Binegar: So we're always think of.
Birne Binegar: Our geometric objects, the basic geometric obviously linear algebra are always going to be the general hyper planes and life simply restricting a number of directions in that hyper plane, we can.
Birne Binegar: Talk on it's essentially the same footing that point lines and planes and even higher dimensional objects.
Birne Binegar: Ah, for playing V1 and V2 need to be linearly independent yes you're getting way ahead of us and we will talk about linear independence until Oh, just before, maybe even after the first mid term.
Birne Binegar: But you're absolutely right, if I had two vectors in this exactly the same direction.
Birne Binegar: Adding them together are never going to be able to move off the common line on which they live, so yeah there is a condition on the factors that if you really want to generate a two dimensional stations you're going to need these to be independent directions.
Birne Binegar: Okay, so.
Birne Binegar: Welcome to the the.
Birne Binegar: That actually becomes a very important question and one that we're going to postpone the exceptional cases for a little while until at least establish our intuition.
Birne Binegar: Okay, so um.
Birne Binegar: Let me close today's lecture by.
Birne Binegar: showing up one way in which these hyper planes are going to show up your linear algebra.
Birne Binegar: So one of the first things we're going to study, aside from establishing.
Birne Binegar: vectors and how to manipulate them one of the first things we're going to look at starting next week is going to be how to solve systems of equations using linear algebra techniques, where, and so, first let me define what a linear equation is so you're a have an equation.
Birne Binegar: But dumb it what makes it linear is the fact that the variables X one X and they all occur in separate terms right, so no X squared X one squared here, nor do I have any term or one variable multiplying another very good they're all separated from each other, like to these plus signs.
Birne Binegar: and
Birne Binegar: These numbers a one through eight in there to represent particular numbers, so let me give you.
Birne Binegar: An example, so one equation one linear equation.
Birne Binegar: In three variables.
Birne Binegar: Two times X one plus x two minus the square root of three x three is equal to five right so.
Birne Binegar: This is an explicit equation I substitute in or the number on the right hand side B and my coefficients to one in the squared minus the square root of three and then X one X two X three oh my variables.
Birne Binegar: The numbers, you know we haven't factor one multiply X two in here we have minus the square three multiply X three those are the coefficients.
Birne Binegar: To one in minus the square three and this number on the right hand side.
Birne Binegar: Not a good name for technical name, for it would be the homogeneous term don't worry so much about nomenclature at this point.
Birne Binegar: It just plays a different role than these coefficients but, in any given the equation, the numbers a one through eight n in the number of be are going to be stated explicitly what's going to be unknown or the values of the variable that actually solve the equation.
Birne Binegar: Alright, so what i'm going to show you now to finish off today's lecture that anytime you have a linear equation, and you have a solution, then the values of the variables X one X and for solution of the equation.
Birne Binegar: formulated as an ordered list is always going to be a hyper plane inside the space of barriers.
Birne Binegar: So so that's the State this is well i'll show you the solutions have a linear equation is always a hyper plane in the vector space of variables, since I have in variables, the each going to.
Birne Binegar: correspond to a particular number, I have a list of in numbers that's going to be n dimensional vector okay so.
Birne Binegar: Let me show you this, the first thing you do is you use a little high school algebra to solve this equation to for the variable X in right, I hope you can readily surmise how like.
Birne Binegar: To call these terms to the right hand side of the equation, and then you buy you buy a in.
Birne Binegar: Alright, so I can at least solve for one of the variables using the equation directly.
Birne Binegar: And then, when I write down a solution vector well this equation says that this vector is actually going to solve the equation, then the last component has to be given by this formula so let's substitute in.
Birne Binegar: Okay, all I did I replace the last component of this vector by his formula coming from the original equation, and then, what I can do is I can.
Birne Binegar: peel off this initial term this constant term be divided by eight again let's just put that in the vector by itself, and then this term and bullet one X one X one also occurs in the first component so sorry, so this should have been sorry there's a typo there.
Birne Binegar: When i'm writing yes X is equal to 00 all the way down last component be divided by n n.
Birne Binegar: And then I have should have X one X one appears from the first component up here.
Birne Binegar: It doesn't appear in any other component until you reach the last component.
Birne Binegar: And then, all of a pure with.
Birne Binegar: men to a n D one divided by Ian and X one.
Birne Binegar: And then X two will appear in a separate vector it'll appear in the second component, and it also will appear in the last component and finally X N minus one is going to appear.
Birne Binegar: Sorry didn't do this point right it's going to be 00, then the second to last component X, to the N minus one times.
Birne Binegar: They N minus one minus sign divided by a an x minus one so always done here is i've taken this vector expression and i've broken it up into a sum of factors.
Birne Binegar: The first factor doesn't involve any of these variables The second factor involved only X one third deck and the only both X two.
Birne Binegar: And then the last step would be to pull out the common factors of X one X in each lovely.
Birne Binegar: Last N minus one terms, and then you see that the solution vector is actually a vector that's constructed.
Birne Binegar: from starting point going a certain distance X one on one direction, a certain direction X two along the different direction dot dot dot plus a certain distance X minus one along the final direction, and so in the.
Birne Binegar: vectors in the solution have the form of a hyper plane of dimension in minus one effectively.
Birne Binegar: Okay, so um.
Birne Binegar: I have a lot more to say about linear equations starting on Wednesday, but that's enough for today so we'll stop here, I hope you enjoy your.
Birne Binegar: Missing can you upload the final missing slide yes, I will I was editing the slide just prior to the election until this last bit is going to be included as soon as I upload it will will be immediately after class okay so um.
Birne Binegar: With that, let me wish you a happy holiday weekend and i'll see you again on Monday and again just wait five or 10 minutes for me to upload the.
Birne Binegar: edited slides I finished them just prior to lecture that's why I was starting to lecture right on time, rather than five minutes early but i'll do that immediately okay.
Birne Binegar: Alright, so have a good weekend.
Vivian Taylor: So for our web assign homework that's do um are we able to do that now, or is, do we need to wait to finish the maitre matrices part.
Birne Binegar: No, you can start those problems now.
Vivian Taylor: Okay, so what.
Birne Binegar: What I intend to do on Wednesday is i'm going to hope, if you have trouble understanding the syntax of the Web assign problems, how to enter vectors etc i'll try to do an example that explicitly in class but.
Birne Binegar: Early first assignment.
Birne Binegar: shouldn't take you much more than 15 minutes to do so don't worry about it, you might have a look at it, to see what you're up against.
Birne Binegar: yeah but don't worry too much about you'll have plenty of time.
Vivian Taylor: Okay, so I remember on Wednesday, you said that sometimes the vectors are listed vertically, instead of the components are listed vertically instead.
Of.
Birne Binegar: that's what I did here so much easier yeah.
Birne Binegar: You see how that I pull apart that vector if I have written them horizontally you, it would take a bunch of lines to see.
Vivian Taylor: Right okay fantastic Thank you.