Birne Binegar: Okay, I guess, we can begin.
Birne Binegar: So first thing to point out.
Birne Binegar: Is our first exam will be this Friday and I brought up the.
Birne Binegar: Courses canvas web page just to show you some materials that that are posted there that.
Birne Binegar: You can use a study eight first of all.
Birne Binegar: let's see Oh, I should also note that homework sets four and five are supposedly should be done by today, so I can answer any questions you have on that.
Birne Binegar: The due date on web assign is like may 7 or something like this because i'm not.
Birne Binegar: On web assign my more or less force do i'm going to accept late homework to push that due date all the way to the end of the Semester otherwise once the due date has passed you can't turn in homework don't i'm.
Birne Binegar: showing you the due date of homework on canvas not on web assign directly, but if you have that done by today that's great because then you you've done with the homework that's likely to be covered on Fridays exam and.
Birne Binegar: The other thing I wanted to point out is um aside from.
Birne Binegar: The lectures so lecture slides and.
Birne Binegar: is at the very bottom let's say two things.
Birne Binegar: I posted solutions to the web assign homework so.
Birne Binegar: Even now, even after you've turned it in you can still look at these solutions and, in fact, the latest homework problem sets four and five, I actually posted the solutions last week.
Birne Binegar: Even though they're do today, because my emphasis is I really want you to do the problems yourself.
Birne Binegar: When you look at the solutions they're probably not the same problems that web assign gives you because web assign always random ICES.
Birne Binegar: The numbers that appear in a problem, but if you're interested in how to solve them, you can look at the sample problem and they'll show you that, basically, the technique it's just a numbers are going to be different, so you should be able to work out homework problems.
Birne Binegar: If you can't do directly, then you should be able to refer to the my solutions and figure out what techniques are actually use or what the.
Birne Binegar: web assign is expecting as an answer you can look at those solutions, but i'm also happy to talk about solutions to homework problems in class as well, so if you have, if you have problems we can maybe address those today and then.
Birne Binegar: At the very bottom of that page, I have some practice that you have relevant ones here are.
Birne Binegar: This mid term one practice exam i'm going to actually go over that examine class, but you might have a look at it beforehand, because they'll show you essentially what the problems are going to look like on the exam.
Birne Binegar: But bring it up.
Birne Binegar: yeah web assign or canvas is not that great over.
Birne Binegar: The campus wi fi.
Birne Binegar: You know it doesn't even.
Birne Binegar: doesn't come up very well.
Birne Binegar: Alright, so i'm just going to abandon this, I think.
Birne Binegar: It anyway i'm going to go over the first problem set on on on or the practice of the among the go over that explicitly on Wednesday.
Birne Binegar: let's see what we got here.
Birne Binegar: Alright don't.
Birne Binegar: Have you run into any any difficulties with the homework.
Birne Binegar: Right, hopefully that's a good time.
Birne Binegar: What i'm gonna do today, then it's i'm going to press forward a little bit with material that well we'll go into more detail, after the first exam.
Birne Binegar: On Friday, I talked a bit about closure under an operation and sub spaces and I be damned to give a few examples of what a subspace is.
Birne Binegar: And so i'm going to talk about that material today, maybe we'll get to this last topic which is bases for some spaces today, but if not.
Birne Binegar: No big loss here your main thing you should be worried about this week is the exam on Friday and i'm just using today's lecture lecture just to get a little bit ahead material that will cover immediately after the exam.
Birne Binegar: Okay, so let's start now so, how does this phrase mathematically, of course.
Birne Binegar: The sets that are we talking about individual vectors in this course, and we talked about sets of vectors like hyper planes or solution sets of linear equations.
Birne Binegar: Actually, those two examples, whether they be hyper planes or sets of equations, even though they come up in linear algebra the most frequent kind of.
Birne Binegar: subsets of vectors that we encounter in this course, or what called sub spaces and is that name indicates, it's a smaller extra space that sitting inside a larger vector space, but in fact it's something even.
Birne Binegar: more restricted to that in this course, we always concentrate on what's going on with scale or multiplication in vector addition maintaining compatibility of.
Birne Binegar: An object or an idea with respect to those two operations that's a key part of the linear algebra and, in fact, when we look at subsets of a vector space.
Birne Binegar: Something it will always been concerned with is hotter those subsets behave with respect to our fundamental operations.
Birne Binegar: So we'll use that language that a subset is closed under scale multiplication if.
Birne Binegar: By applying scale and multiplication you stay inside the set so here, I have a number a vector that starts inside the subset and then, what I want, in order to be closed on the scale of multiplication is that, after performing this HALO multiplication I dumped the part that set.
Birne Binegar: So, for example.
Birne Binegar: here's a counterexample you have point on a unit circle.
Birne Binegar: every coin on the circles that distance one from the origin, but if I say I take a vector and I multiply it like to call that be this might be to be.
Birne Binegar: That scale multiplication has taken me all circle until a circle is going to be an example to set that is not closed under steel and multiplication because simply making the vector longer takes you outside to set.
Birne Binegar: All right, and then we have a similar notion for the other fundamental operation subset is closed, under a vector addition.
Birne Binegar: If you start off in the set and you add two vectors live in the set that if you stay inside the set under those combination, then you just closed on their.
Birne Binegar: vector edition tradition and here another example of a sentence not closed would be the circle, I have the vector with coordinates 01 and the vector with coordinates one zero of those two point.
Birne Binegar: would be the tips of those vectors by add these two vectors together.
Birne Binegar: I get a point with coordinates one one and, of course, the link to this factor is going to be the squirted to write take the dot product of the spectrum of itself, the one plus one.
Birne Binegar: dot product vector of itself it's a square this link, so, in fact, these two vectors are going to add together and they're going to produce a vector that actually is off the circle it's linked it's going to be the square room too much rather than one so.
Birne Binegar: The circle, even a simple geometric optics it is it's an example of a subset of a set of all two dimensional vectors that's not closed under scale multiplication, nor is it closed under vector addition.
Birne Binegar: Alright, so.
Birne Binegar: Well, the circle doesn't look very linear so you know.
Birne Binegar: it's round, rather than straight, so you might expect that maybe it shouldn't be used in linear algebra and, in fact, there are other sets.
Birne Binegar: That we've considered that looks more like vector spaces, like a line in a plane, you can think of this if you just.
Birne Binegar: myopically focused on the way these points are arranged next to each other, look short like a one dimensional vector space because simply by.
Birne Binegar: During a single parameter, you can move along this line, but it's going to turn out that even.
Birne Binegar: gen hyper planes, whether it lives in a plane or three dimensional hyper planes embedded in a 10 dimensional space they're not going to be sub spaces or they're not going to be closed under Taylor multiplication in depth tradition in general.
Birne Binegar: Another way that.
Birne Binegar: sort of subsets that we considered the way they.
Birne Binegar: miraculously appeared in linear algebra is when we try to solve systems of linear equations we get solution sets that are, in general, hyper planes, but again, even those solving a system of linear equations is right.
Birne Binegar: In the.
Birne Binegar: Natural arena of linear algebra but turns out that the solution sets of linear equations or in general, not closed under scale or multiplication or vector addition.
Birne Binegar: let's see where's.
Birne Binegar: Particularly after the first exam it's going to be on the special subsets of the vector space that are closed with respect to scale or multiplication and vector addition.
Birne Binegar: meaning if you start in the set and you scale a multiply you're still in the set if you have two factors that start off and so after you add them together, you still inside that subset.
Birne Binegar: If both these properties hold then we say we have a subspace of our end and here, this is going to be.
Birne Binegar: Some places I aren't that going to be hyper planes but they're going to be special hyper planes.
Birne Binegar: And so, but as the name suggests, substations you should think of this little vector space is sitting inside larger vector space, but not only are they going to look like.
Birne Binegar: Little vector spaces sitting inside a larger vector space, the key part is that they behave well respect to the two fundamental vector space operations.
Birne Binegar: Alright, so i've gone over this unit circle example whether you scale a multiply vector vector on the circle, or you add two vectors on the circle, it takes you off the circle So those are not safe spaces.
Birne Binegar: here's.
Birne Binegar: A set that one will construct by.
Birne Binegar: Looking at the point in the plane, but requiring the individual components to be integers.
Birne Binegar: Then this subset it's going to be closed under a vector addition because take any vector in this set it's going to be.
Birne Binegar: correspond to a pair of integers in each case I add those things together what am I get well an integer plus an integer is going to be an integer so.
Birne Binegar: Even after adding these two actors, I still end up with a vector that satisfy this condition it's going to have two components and each component is still going to be an integer so this set key is closed under vector edition.
Birne Binegar: But it's not a subspace the reason be is that while it's closed under the vector addition it's not closed under scale or multiplication see that.
Birne Binegar: being closed on the scale of multiplication meet any any vector inside the set times any number has to stay inside that same set.
Birne Binegar: But if I multiply a vector that lives inside this set to both components are integer so it isn't T, but if once I multiply by the square to to I get a vector he's component first component is not an integer so it doesn't live inside that set.
Birne Binegar: Alright, so that means, while this T is maybe halfway towards the subspace because it's not closed under scale and multiplication it's not going to be a subspace.
Birne Binegar: here's yet another example here's a hyper plan, this is a set of vectors that we create by choosing a starting point, p zero.
Birne Binegar: A set of directions w want to wk and the way to set is constructed as you start at the point, and then you go an arbitrary distance long any of these K directions.
Birne Binegar: You let these parameters s one through it okay very over the entire rely, and what you get is something that looks like a.
Birne Binegar: Effectively, a K dimensional vector space, you have Cape Cape parameters here, but in fact it's going to be sitting inside a larger vector space in general, so this might be our 10 and maybe I only use five vectors.
Birne Binegar: So that is some kind of mixed object right in some sense every vector here is going to live in the original vector space, but.
Birne Binegar: When you look at the dimension of the thing that's maybe depicted by the number of free parameters, you need to describe this object well it's only going to depend on K parameters, so you have effectively a katie mentioned a subset of vectors sitting inside a larger vector space.
Birne Binegar: And it turns out that hyper planes in general are not stop spaces, even though we just use linear combination to construct them.
Birne Binegar: The stuff spaces are not closed under vector addition or scale or multiplication.
Birne Binegar: it's easiest to show that this cannot be subspace because it does not it doesn't contain the zero vector you won't be a subspace.
Birne Binegar: So assume that the set does not contain the zero vector.
Birne Binegar: Then.
Birne Binegar: If you multiply any vector inside the set by the number zero well that's going to produce the zero vector because zero is going to hit each component at the zero times anything produces zero.
Birne Binegar: So after you do the scale of multiplication by zero you're going to get the zero vector but if I said he happens didn't somehow missed the origin, then.
Birne Binegar: The zero vector is not going to live in scientists said here's a picture of that situation here's a effectively a one dimensional hyper plane on the two dimensional blackboard but because it doesn't contain the origin.
Birne Binegar: it's not going to be closed on the scale of multiplication.
Birne Binegar: I take any vector along here and I multiply by zero I get I move off that line and I go to the origin, but the origin like if.
Birne Binegar: We had i've drawn this picture is definitely not a point on this line and so scale a multiplication of any point on this line by the number zero it's going to take you outside that set.
Birne Binegar: And so that's that's really common know.
Birne Binegar: The real the real problem here is this point P zero.
Birne Binegar: What you the way.
Birne Binegar: We would actually construct the line.
Birne Binegar: Is we would take a point P zero and we'd add to it, some direction and allow.
Birne Binegar: Self the stray from the point P zero Bytes and arbitrary distance so here's the p zero maybe this is to direction D and then by changing the scale or multiple of the I can move.
Birne Binegar: backwards, or if I multiply by negative number, I can move upwards and off to the left and i'll fill out the various points on this line.
Birne Binegar: But it's this fact that we're offset from the origin that's what's actually causing the problem, because once you will offset in this hyper playing does not contain the origin.
Birne Binegar: i'm unable to multiply by zero without leaving the set top hyper planes in general and not going to be sub spaces, even though it looks like we construct them in a natural way by just taking linear combinations of factors.
Birne Binegar: All right, and in fact.
Birne Binegar: This property that if you anytime you have a subset of a vector space that does not contain the origin, you have the same problem.
Birne Binegar: If your subset doesn't contain the origin, then scale multiplication by zero is going to take you outside the set and you're not going to have the subspace, so this is a.
Birne Binegar: easy thing to remember oftentimes you can identify us.
Birne Binegar: a subset of a vector space is not being a sub space simply by the fact it doesn't contain the origin.
Birne Binegar: Right, so let me.
Birne Binegar: Now, look at us.
Birne Binegar: A case the hyper plane that's always a subspace.
Birne Binegar: So this has been a special case of a pipeline what we'll do is we'll start off with a set of factors and.
Birne Binegar: we'll take all possible linear combinations of that given list of vectors don't just take a linear combination let those coefficients range over the real line notice the difference here between the span of vectors its definition and the hyper plane before.
Birne Binegar: The hyper plane before we have this offset and it's actually the source of all why this hyper plane is not a subspace.
Birne Binegar: When the offset doesn't actually occur, then in fact a span of a set of vectors set of all possible vectors you can get by simply taking linear combinations have you given list of factors that's always going to be a subspace.
Birne Binegar: Let me show you here's here's the fan remember it's defined as the set of all possible linear combinations let's take a number any number could be zero could be square root of two and let's multiply a time so typical vector sitting inside this set.
Birne Binegar: Well, we do the scale or multiplication and what happens upon multiplying by Lambda.
Birne Binegar: I get a vector, which is a number of times w one and number times w two dot dot dot a number of times the last factor and so.
Birne Binegar: The result of scale and multiplying a given vector by any number Lambda produces a vector of this form just a bunch of numbers times the given list of vectors summed up so that is closed under scale or multiplication the set.
Birne Binegar: and, similarly, if I take two factors that live inside the span of US given list of factors so they're each going to be formed by taking linear combinations of your given list of actors.
Birne Binegar: If I add those together well just add them up and keep the common factors here, these common vector factors and you see that the one plus the two is going to be s one plus one time study you want.
Birne Binegar: To plus T two times w two etc, and what I end up with is.
Birne Binegar: A linear combination of the original set of vectors with.
Birne Binegar: Simple real numbers as as coefficients and so.
Birne Binegar: This factor is of this form it's a bunch of numbers time to given list of vector summed up.
Birne Binegar: So.
Birne Binegar: These separate you formed by taking the set of all possible linear combination of factors is always closed under scale of multiplication in depth tradition, so it will be a subspace.
Birne Binegar: All right, let me.
Birne Binegar: that's the first.
Birne Binegar: First, fundamental example.
Birne Binegar: A span of a given list of actors.
Birne Binegar: w.
Birne Binegar: wk.
Birne Binegar: let's write it out explicitly so we know exactly what we're talking about we're looking at every actor, we can create by taking certain parameters times are given list of actors.
Birne Binegar: OK.
Birne Binegar: OK, and then regenerate an infinite set of vectors by letting these parameters.
Birne Binegar: range over the entire real life.
Birne Binegar: Any sets of this form or always sub spaces, because they're closed under scale of multiplication and vector edition is that was that offset P zero that prevented the general hyper plane from being a subspace.
Birne Binegar: Alright, so.
Birne Binegar: here's another example, and of course it's related to.
Birne Binegar: Solutions of linear systems are course hyper planes That was our one of our early discoveries in this course, and so you should expect since the solution set.
Birne Binegar: up a linear system is always a hyper planes and general hyper planes are not safe spaces, you should expect that a set like this shouldn't it's probably not going to be a subspace in general, let me show you that true.
Birne Binegar: Suppose we have the solution, set up a linear system.
Birne Binegar: question is is the set closed under scale and multiplication well what I mean by that is if I start off with a solution.
Birne Binegar: Of this equation so belong to the set so what happens when I scale and multiply my solution.
Birne Binegar: By a number Lambda is it still a solution well let's see what equation that satisfies a times Lambda times, why well that scaling multiplication can be delayed until after we do the matrix multiplication but a why it's supposed to be D, so I just have Lambda times B, but now.
Birne Binegar: it's not that Lambda, why does he satisfying equation, it does it just doesn't satisfy the same equation, which started off with a time Lambda y equals Lambda be rather than be, and so the set.
Birne Binegar: Is not.
Birne Binegar: it's not closed under scale multiplication because a solution.
Birne Binegar: After scaling multiplication satisfies a different equation, the equation, we started off with.
Birne Binegar: Nor is it close Center vector addition either to post, we take to solution, why and w that both satisfied this equation add them together, and the question is, does this summit vectors y plus w does it still satisfy the same equation.
Birne Binegar: Well, when you distribute that matrix product over the vector some a y is equal to be because, why is a solution wc solution, so a w is equal to be and we get to be rather than be.
Birne Binegar: So, because the vector some when I plus w that's not satisfied the same equation is y and w this set is not closed under vector edition either.
Birne Binegar: Alright So here we have a set Curry naturally in this course, the solution, set up a linear system and it turns out not to be a subspace because it's not automatically closed under either steal a multiplication or vector addition.
Birne Binegar: Now, remember.
Birne Binegar: For sub spaces.
Birne Binegar: Well, for hyper planes, not every hyper plane, with the subspace but as long as we included the origin, then we.
Birne Binegar: Then we would have a subspace well i'm going to do a simple something similar here it's true that.
Birne Binegar: solution sets of linear equations are not generally sub spaces, but for special systems of linear equations they will always be sub spaces and the condition is.
Birne Binegar: Effectively, that what you want, on the right hand side of your equation is not a non zero vector be, but if you have a zero vector on the right hand side, then it's always going to be closed under the solution set is always going to be closed under scale and multiplication effect tradition.
Birne Binegar: So let me say.
Birne Binegar: you're on this blackboard let me put the things that are not sub spaces and you'll start to see the pattern.
Birne Binegar: Plus.
Birne Binegar: tk.
Birne Binegar: wk.
Birne Binegar: letting those parameters.
Birne Binegar: range over the title line.
Birne Binegar: Okay, so these.
Birne Binegar: are not going to be sub spaces, but these hyper planes were P zero that starting point is effectively the origin and insert the zero vector there into the some it's not going to change the result, and all that is going to be a subspace and what i'm going to show you now.
Birne Binegar: that a solution set.
Birne Binegar: up a linear system.
Birne Binegar: we're on the right hand side, instead of having a.
Birne Binegar: Non zero back do we have the zero vector that's also going to be a subspace but.
Birne Binegar: In contrast to that the solution set up.
Birne Binegar: A.
Birne Binegar: equals B is not a sub space.
Birne Binegar: in general.
Birne Binegar: happen if this non zero term here is going to screw things up.
Birne Binegar: Let me show you how that works so let's first consider what.
Birne Binegar: i've already shown you that solution sets up.
Birne Binegar: What i'll call in homogeneous linear system for the right hand side it's not the zero vector these things are not closed under scale and multiplication or vector edition.
Birne Binegar: But so long as the right hand side of the equation is equal to zero, what happens when you add two solutions is you end up adding eight times y equals 08 times w is going to equal 00 plus zero is equal to zero, so you start off with two solutions of.
Birne Binegar: This equation you add them together, you still have a solution.
Birne Binegar: and
Birne Binegar: Furthermore, if you take a solution of this equation, and you multiply by a number, you can delay that scale and multiplication till after you've applied the matrix a to your vector why.
Birne Binegar: But a times, why would be zero and scale or multiple of the zero vector is always zero vector both scaling multiple solutions of this or vector sums of solutions of this they stay inside the solution set and so.
Birne Binegar: Each example here is a special case of either a hyper plane or linear system but it's only in the special cases that we actually.
Birne Binegar: get a subspace.
Birne Binegar: Okay, and then you should expect some kind of analogy between these two situations because, after all, when you solve linear systems, you always get hyper planes has solutions, so the fact that.
Birne Binegar: Most hyper planes are not going to be up spaces that's reflective of the fact that most linear systems are not going to have sub spaces as their solution set on the other hand.
Birne Binegar: There are special hyper planes that are always sub spaces in fact they're matched by special systems of equations that are always substations who solution sets are always sub spaces.
Birne Binegar: Okay, so now.
Birne Binegar: What we have so far is the definition of the subspace be a subset sitting inside a set of back you're sitting inside a larger set of factors.
Birne Binegar: Such that closed under scale multiplication in that tradition, and you will have two basic prototypes for making such guys one is take a list of factors.
Birne Binegar: And just take all possible linear combinations.
Birne Binegar: And the second way.
Birne Binegar: of creating a subspace.
Birne Binegar: is to just.
Birne Binegar: Take a coefficient matrix a and then look at the solution, set up a time so vector X is equal to zero.
Birne Binegar: Okay, so.
Birne Binegar: What i'm going to do to finish off today's lecture.
Birne Binegar: Is i'm going to concentrate on this first example introduce some.
Birne Binegar: general problems we're going to run into as soon as we start to try to manipulate.
Birne Binegar: Sub spaces.
Birne Binegar: several problems.
Birne Binegar: That, I want to try to address, maybe the first one would be.
Birne Binegar: subspace is.
Equal.
Birne Binegar: let's take it as a span of a set of vectors i'll call them T.
Birne Binegar: Plus, plus parameter tk can't be accurate bk let the parameters key one through tk very over the entire real life.
Birne Binegar: That is the span of the vectors D one through the UK and it is a subspace what, if I had another.
Birne Binegar: Similar stuff said.
Birne Binegar: Maybe s one w one.
Birne Binegar: Plus.
Birne Binegar: Yes, L.
Birne Binegar: w El.
Birne Binegar: He won sorry that's one to sell.
Birne Binegar: All real numbers, I have the spans of two different list of factors.
Birne Binegar: How do I know.
Birne Binegar: If they're the same set of points that's included in each of these sets, how do I know whether it's the same or not not easy to tell certainly not easy to tell directly from.
Birne Binegar: The way i've set up the sets.
Birne Binegar: So that's one question.
Birne Binegar: But a second question.
Birne Binegar: That dumb.
Birne Binegar: That we're gonna have to address.
Birne Binegar: fairly quickly is.
Birne Binegar: Well it's sort of an issue when, in the way we've set up the set.
Birne Binegar: So if we let me give you an explicit example suppose the one.
Birne Binegar: is equal to one zero expose the two is equal to one one and maybe the three is equal to 011.
Birne Binegar: Then.
Birne Binegar: If I take the span of this these factors.
Birne Binegar: span of the want the to MP3 well what is going to consist of it's all the vectors I can get by now.
Birne Binegar: Oh wow P one P one P two.
Birne Binegar: The two plus 233.
Birne Binegar: But what i'm trying to detect here is that these guys are all living.
Birne Binegar: In a two dimensional space.
Birne Binegar: But if I look at this said it looks like it depends on three parameters key one tier two and tier three so is this a.
Birne Binegar: This is a three dimensional space sitting inside a two dimensional space clearly something's wrong there because you have a larger objects trying to squeeze it into a.
Birne Binegar: Just a two dimensional space so something's wrong here is that I want to describe point in this subspace he won T to MP3 or not good coordinates for the point right, even though once I choose.
Birne Binegar: To one equal to two T two weeks equal to three and three is equal to or I get a specific vector, and so I can just I can get every point in this space like choosing particular values of numbers, unfortunately I cannot tell.
Birne Binegar: You whether two vectors or even different if I use this construction to actually write them down.
Birne Binegar: I think, on the slides, I have a.
Birne Binegar: clear example of that.
Birne Binegar: But first some nomenclature, so when we.
Birne Binegar: create a sub space of vectors by taking all possible linear combinations with a good list of factors we say that s is the subspace generated by the factors in the list, and we also say that the individual vectors do you want DK our generators for the subspace.
Birne Binegar: So what I want to show you in this next slide is I guess close to this example, I have a set of three doctors.
Birne Binegar: And I can think of writing down examples of vectors that live in as simply by choosing a particular number C one C 23 so let me see note, like the subscript C one C two C three the vector and S that I formed by using particular values for each of these coefficients.
Birne Binegar: When I want to show you is that I use one set of values 110 and I construct the corresponding vector my using the corresponding linear combination.
Birne Binegar: One times the first director one times the second factor zero time third vector I get a particular vector like take a different set of numbers C one C two and see three, I have a different linear combination to compute but upon computation you see you get the same vector.
Birne Binegar: So we have a problem here.
Birne Binegar: Even though I can create all the vectors in the subspace like choosing particular numbers.
Birne Binegar: Unfortunately I don't really have an easy way of detecting whether one set of numbers and another set of numbers do they produce the same factor, or are they different vectors.
Birne Binegar: We have a problem of coordinate icing vectors inside the subspace, even though we have a way of writing them down it's hard to tell.
Birne Binegar: Simply from looking at these expansions before we compute it's hard to tell whether this vector is the same as this, we have to actually do the computation and then do the comparison.
Birne Binegar: Well, this is going to make the problem of determining whether to sub spaces are the same kind of difficult not only could we have different lists of vectors creating the same some space, but even if we work from the same list of vectors there may be multiple ways of representing.
Birne Binegar: A unique vector inside the subspace here, I have two different choices up coefficients that lead me to the same vector, and so my coordinates if you think of.
Birne Binegar: Try to interpret C one C two and C three s coordinates of the vectors inside the set yes they're not very good coordinates because.
Birne Binegar: If you change your coordinates you don't necessarily change the point that they they correspond to.
Birne Binegar: Alright, so what this is leading up to and what will resolve after the first exam if the notion of a special set of vectors that generates a subspace with all basis for the subspace these are going to be.
Birne Binegar: directors that are sufficient so that we can reach every point in their subspace simply by taking linear combinations but we're going to put one extra condition on the US.
Birne Binegar: What we want is that anytime we choose a vector not only can we represent it as a linear combination of you given list, but we want it to be uniquely.
Birne Binegar: Once you fix the choice of see one two ck you get a unique Dr not liking this proceeding example where I can use two different choices of coefficients and get the same vector requirement we're going to impose.
Birne Binegar: For the vectors.
Birne Binegar: Of a basis it's going to be that anytime you write a vector inside the subspace you expand it in terms of the basis vectors, then the choice of coefficients he wants you to up the cta that's going to be unique.
Birne Binegar: If this is true what that's going to allow us is to coordinate is a point of points or vectors that live in a subspace we can use these coefficients as their coordinates.
Birne Binegar: So.
Birne Binegar: here's effectively the definition of what a basis is it's a set of factors that sufficient.
Birne Binegar: You haven't sufficiently many directions, so you can reach every point your subspace simply by taking linear combinations but in addition to being able to reach every point you want to do so in a way that if you have.
Birne Binegar: Two different linear combinations that produce the same vectors what you want, is that necessarily the coefficients have to be exactly the same.
Birne Binegar: And so, if this is true, that means anytime I want to focus on a particular vector there's only going to be one set of coefficients that will produce that vector be of these linear combinations, and so I can use these numbers see one two ck to specify uniquely the point in the subspace.
Birne Binegar: That helps me address that the additional problem as well.
Birne Binegar: The vectors that generated subspace they're going to be up dimension, the same size as this is the space they live in, so if I had a list of 510 dimensional vectors then.
Birne Binegar: The span of the vectors is going to be a subset of attend dimensional vector space, because it made from 10 dimensional vectors.
Birne Binegar: But the dimension of that that said that's really best be described in terms of these parameters, because I only need K parameters to reach every point in that katie mentioned subspace.
Birne Binegar: And so.
Birne Binegar: The purpose of the basis is it actually not only will tell us.
Birne Binegar: How do you reach.
Birne Binegar: How to reach every point in your sub base uniquely but also give us coordinates that is uniquely described every point in your subspace.
Birne Binegar: Okay, so.
Birne Binegar: To be to be a basis.
Birne Binegar: What you need is i'm sorry that type of there, but what you need is a list of vectors such that you can reach every point in your sub say simply by taking linear combination and you want every point in your subspace.
Birne Binegar: to correspond to a unique set of coefficients.
Birne Binegar: That way.
Birne Binegar: As 10 dimensional vectors.
Birne Binegar: Then I said I set up a space.
Birne Binegar: i'll just take it to be the span of these vectors.
Birne Binegar: V1 V2.
Birne Binegar: Three well.
Birne Binegar: Since i'm taking linear combination to 10 dimensional vectors whatever that set is it's going to be a subset of our 10.
Birne Binegar: But if.
Birne Binegar: If the set of vectors V1 V2 P three.
Birne Binegar: Is a basis.
Birne Binegar: For this set.
Birne Binegar: Then.
Birne Binegar: Every vector is sitting inside the set.
Birne Binegar: can be uniquely written.
Birne Binegar: As the let's see.
Birne Binegar: Yes, equals.
Birne Binegar: c One the one plus C to B to C three.
Birne Binegar: The three, and in fact from this.
Birne Binegar: I can form an ordered list.
Birne Binegar: I just grabbing the numbers, I need to create a given point and that's.
Birne Binegar: A list of three numbers, this is how you see.
Birne Binegar: The dimensionality of the subspace it's not this the dimension of the space in which all its elements live rather it's the number of parameters, you need to reach every point in this space so.
Birne Binegar: that's, the main purpose of introducing this notion of faces, not only is going to help us decide.
Birne Binegar: Being able to give good coordinates for every point inside our subspace.
Birne Binegar: it's also going to be able to give us a more accurate determination of what the size of that subspace is.
Birne Binegar: If this.
Birne Binegar: In our first example I gave you a series of three vectors in aside a two dimensional space that I could write.
Birne Binegar: Like might consider the span, I have three parameters, I could change, but what was bad about that example is that I found two different choices for tier one tier two and tier three that led me to the same vector so these.
Birne Binegar: This set of generators did not give me that good natural coordinates because there was more than one way of writing a given vector and site is set.
Birne Binegar: As a linear combination like given list what's special about a basis, I its definition is that.
Birne Binegar: Every vector inside this set can only bit be written one way as a linear combination to get a list.
Birne Binegar: By the uniqueness of that I can use the number C one C two and see three to uniquely specify every element of the set.
Birne Binegar: Since I only need three parameters to describe every element of the set that's why regarded as a three dimensional subspace even though its individual elements are living in a 10 dimensional space, the set itself only has three degrees of freedom.
Birne Binegar: Okay, so.
Birne Binegar: i'll stop here today, and then.
Birne Binegar: After the exam we'll come back we'll look at substations now begin to address the more complicated issues like, how do you tell when there are two sub spaces are the same or not.