Birne Binegar: Okay.
Birne Binegar: we'll get started now so today.
Birne Binegar: For homework said six supposedly done today, of course, the due date on web design that the very end of Semester, the due dates that I post on canvas.
Birne Binegar: Or the dates.
Birne Binegar: The client get the homework done so, the steak on the class homework that says, how do we make you're seeing their faces out of fine bases for those the calculations are done i'm not going to be relevant, this week, so I want you to try and get those done by night.
Birne Binegar: Wednesday through Friday like should you start socially at face value nation don't know how to find their basics.
Birne Binegar: So, and then i've also to do homework homework assignment seven that's going to be viewed the Monday after spring break or not next Monday, the following day.
Birne Binegar: And then.
Birne Binegar: The second exam it's probably going to occur, the week that can come back from spring break I think we'll get the linear transformation this week.
Birne Binegar: I would think that maybe Wednesday of.
Birne Binegar: The stage seven so that would be one that Wednesday the 23rd of March that's a likely date for the second exam but will actually scheduled.
Birne Binegar: Maybe on Friday, this week, when I have a better idea of how far we've gotten your transformation through material on linear transformation by Friday doesn't make sense talk about the homework and do a review session on Monday and then have an exam on Wednesday.
Birne Binegar: could also have it on Friday.
Birne Binegar: which would be the 25th of March, however i've already got one requested down someone's going to be out of town, the other one has a big jam on that day, so i've had to request not to have it on Friday, the 25th so if we don't do it on the 23rd then it's possibly would get pushed or two.
Birne Binegar: But we have we can come back from spring break, but if we do that, then i'm going to start talking about new material us before we have the damn sure look up to these information.
Birne Binegar: Okay, so that's that's where we are in the course today want to.
Birne Binegar: quickly review the notion of linear transformation main topic for today's we live linear transformation data.
Birne Binegar: Is we're done so often is course i'll introduce an abstract concept, like a base stuff space and then i'll try to give you a calculation of platform, you can compute.
Birne Binegar: And that will typically involve make you'll see the same thing that and the Faculty to some places that we can tap your transformation, where we actually.
Birne Binegar: display them explicitly by looking at the quarter funny matrix matrix recorded on your transformation.
Birne Binegar: So let me put some.
Birne Binegar: More ideas of that we've been dealing with.
Birne Binegar: Since.
Birne Binegar: Since we came back from the first exam.
Birne Binegar: in mathematics.
Birne Binegar: mathematics actually starts with.
Birne Binegar: To really basic off set and functions don't we.
Birne Binegar: Like.
Birne Binegar: We came back from the first exam.
Birne Binegar: Our topic principle, it was a special case of the idea of a set of objects that were the so called substances.
Birne Binegar: vector space aren't am now what these were where they're special subsets say yes every every element of so it's also an element of art and that just saying it's a subset, but what we require in addition.
Birne Binegar: We demand since we're linear algebra compatibility of this notion, with respect to scale or multiplication in vector nation, so we would demand that you start off with number.
Birne Binegar: And you start off with a vector.
Birne Binegar: That lives inside this, that is.
Birne Binegar: Not necessarily a scale multiple of your vector by the number stays inside that set.
Birne Binegar: Second, of all if you have two factors say be w that start off living in tighter subset that after you add them together.
Birne Binegar: you're still inside that self specs.
Birne Binegar: Well that's a.
Birne Binegar: Special says that are relevant to linear algebra.
Birne Binegar: So Emily what is, we discovered last time.
Birne Binegar: For linear algebra we have a special case for function.
Birne Binegar: deal and linear out for constantly with vector space is work we focus on functions between factor.
Birne Binegar: We aren't in.
Birne Binegar: Our again and i'm not list incident so much in every possible function between two vectors pitches but rather ones that behave well we're back to our spec.
Birne Binegar: vector operations so such that.
Birne Binegar: This function has the property that I have a number.
Birne Binegar: And I had a vector that lives inside that domain.
Birne Binegar: vector inside.
Birne Binegar: So you got your spaces, where the.
Birne Binegar: function begins, then.
Birne Binegar: Then compatibility requirement is that he applied to a scale or multiple of the has to be the same thing as that scalable patient.
Birne Binegar: client to the image of me, so what this is saying is I can do the scale of multiplication before or after I apply the linear transformation that I get the same result well that's not going to happen for every function, but you must be the case for linear.
Birne Binegar: And second of all if we start off with two vectors V1.
Birne Binegar: And V2 that start off living in time to domain, then what we require is that.
Birne Binegar: We add them together.
Birne Binegar: We get the same result as if we delayed that addition until after we find linear transformation.
Birne Binegar: Okay, so.
Birne Binegar: What the way to think about this is that the mathematically we're interested in sets and then later more advanced mathematics functions between set that will allow us to relate different sex.
Birne Binegar: What will require in linear algebra is even more, we want these sex and functions to be paid well respect to our fundamental factors faith offer a constant payments.
Birne Binegar: And so what I did last time.
Birne Binegar: I interviewed linear transformation and as exactly function for this country between two factor space that are compatible scalable patients that tradition.
Birne Binegar: makers ease and then we'll start with talk about some spaces associated.
Birne Binegar: With the function.
Birne Binegar: Okay, so.
Birne Binegar: i'm just putting up the definition again of a linear transformation.
Birne Binegar: behavior elemental technical operations that substance of that nation.
Birne Binegar: And I showed you last time there's one.
Birne Binegar: One way of making such a function is to use a matrix I know what we're probably have to use to define a function that that's one factor state to another is make kitchen.
Birne Binegar: Wait on the work is that we start off with a list of numbers, we think about as a column vector we think of exit the column vector then it can be multiplied by a matrix and in fact dimension, which are going to all work out.
Birne Binegar: to write them down.
Birne Binegar: You start off with.
Birne Binegar: A.
Birne Binegar: matrix.
Birne Binegar: And you set X equal to.
Birne Binegar: em one column vector.
Birne Binegar: Then a times X well what this is going to be it's an n by n matrix multiplying by one column vector that's going to produce.
Birne Binegar: We got the match of the inner dimensions which is necessary for this matrix will.
Birne Binegar: result that is going to be in one column vector which you can think of.
Birne Binegar: As an n dimensional vector because in this column there's going to be exactly in entries.
Birne Binegar: Right and then So what does this matrix multiplication do it takes an n dimensional vector.
Birne Binegar: And it produces an n dimensional vector So what is the function between these two vectors.
Birne Binegar: And, moreover.
Birne Binegar: These two properties are going to follow, naturally.
Birne Binegar: Let me show you that real quickly.
Birne Binegar: P subgroup a.
Birne Binegar: So.
Birne Binegar: it's a.
Birne Binegar: matrix.
Birne Binegar: Then I have a function from rm.
Birne Binegar: are defined by.
Birne Binegar: He subscript a X.
Birne Binegar: is equal to a time.
Birne Binegar: Where I interpret this as a column back here that we multiply that this matrix and will produce an n dimensional vector.
Birne Binegar: No, no.
Birne Binegar: Right, so this shows you anytime I have a matrix I can create a linear transformation what our main point of today's lecture is that.
Birne Binegar: The opposite is also true given a linear transformation.
Birne Binegar: same T and Sprint we think of this as a general linear transformation between two vectors basis.
Birne Binegar: Then they'll be.
Birne Binegar: there'll be an associated make.
Birne Binegar: which in some ways it's kind of like the utility of the film about the stage is kind of like the way we use elementary me.
Birne Binegar: What will you tell me to make friends with monitor away implementing.
Birne Binegar: Elementary row operation but doing it via matrix.
Birne Binegar: That actually the goal of today's lecture is to given a function between two vector space associate to that a matrix that will do the same thing and then all our calculations can be done on that concrete miracle object the matrix rather than this more abstract function.
Birne Binegar: So.
Birne Binegar: So that's that's the goal dead not only given a matrix can we create a linear transformation, but given the linear transformation great a corresponding matrix.
Birne Binegar: Well here's the thing everyone your confirmation there's going to be a matrix that the action of acting.
Birne Binegar: Whatever this function does to an element of our am that can be replicated by you do make up for the make.
Birne Binegar: And i'll do it for you, for that matter, and just a couple minutes.
Birne Binegar: But what I want to do.
Birne Binegar: This describe how we get this matrix i'm not looking a little more closely at this matrix multiplication first.
Birne Binegar: So let's take.
Birne Binegar: An MIT and metrics in a column your.
Birne Binegar: And i'm going to write.
Birne Binegar: i'm sorry this elaborate location column subscript J s to represent the jade column of a.
Birne Binegar: More specifically.
Birne Binegar: If we have.
Birne Binegar: A calling.
Birne Binegar: 1812 at 1am similar roads and can we can throw a 2am what I mean by.
Birne Binegar: column J K is the column vector I create by grabbing.
Birne Binegar: The jade call, but this somewhere there's a call on the road indices are going to go from one end but would fix the column index constantly J.
Birne Binegar: Alright, so that's you just look at the matrix can you grab the corresponding call alright So what does that do for us.
Birne Binegar: Well, my claim is that when I will find an arbitrary vector X times matrix a that result of that vocation can be interpreted as a linear combination of the columns of the matrix ED.
Birne Binegar: And in fact well, this is only your combination once numbers time that something up in the fight is is that you compute this matrix product and i've turned to way up to.
Birne Binegar: The entries and the specter and use those as coefficient of the linear combination, where the corresponding factors are the columns of the matrix so.
Birne Binegar: It doesn't take much to prove this, all we have to do is compute this product, here it is and we do this, we dot product and each row with the first factor is the one and only all.
Birne Binegar: produces this n dimensional vector is each row here in a big adopt products can produce the numbers of putting one of the blocks of me mentioning result.
Birne Binegar: And then, what I can do we're familiar with this operate connect operation is basically have a bunch of factors that are being added together and.
Birne Binegar: have them are being multiplied by the parameters X one hey what simply take this n dimensional vector and all expanded respect to the parameter X one X two yeah.
Birne Binegar: So the first set of terms that just X one time the first column a the second set of term have a common factor X two and multiplying the second column with me going back.
Birne Binegar: We did exactly that expansion anytime you multiply a column vector by matrix is equivalent to constructing a linear combination of the call that a where the coefficient or exactly the entries of it out the vector X.
Birne Binegar: Alright well that's first half of what i'm going to tell you the second half is even easier just be familiar um.
Birne Binegar: there's a notion of a standard paces for our.
Birne Binegar: First AC factors will we have one in the first slot and zero.
Birne Binegar: seconds databases doctors work in the second slot we have one zero everywhere else, and he just yeah a bunch of factors, simply by putting one in one of the components of any me mentor better.
Birne Binegar: Alright, so what is special about it well, you get a list of actors and back.
Birne Binegar: The empire state right because.
Birne Binegar: When we see that.
Birne Binegar: next one.
Birne Binegar: let's write X one.
Birne Binegar: X two X two.
Birne Binegar: X yeah.
Birne Binegar: If I write this out, I get X 1100 plus x two times 010.
Birne Binegar: Excellent excellent.
Birne Binegar: Euro euro.
Birne Binegar: 01 and I take this linear combination and I add them all up well the X one only contributes to the first component, the act to only contributes to the second component as well, so this is just.
Birne Binegar: X one.
Birne Binegar: X, so this ordered list of em numbers can be expanded.
Birne Binegar: In terms of the standard basis.
Birne Binegar: Alright, so that's that's code for standard, this is huge this construction all the time, except they tend to call the standard a factory I J K for three minutes.
Birne Binegar: While we're considering more gentle faces all you subscription rather than different time Latin letters in the face to face the same idea.
Birne Binegar: So one thing is that every doctor inside or any written a linear combination, but it's also pretty clear that these guys are linearly and.
Birne Binegar: Because the only way a linear combination of the standard basis vectors can produce the zero vector is well what kind of happened here, you said zero.
Birne Binegar: equal to this linear combination well this linear combination is equivalent to this vector and this factor is not equal to zero vector unless all the components are equal zero.
Birne Binegar: But then, if you have a set of factors such as the only linear combination that proves zero vector is the one we're all.
Birne Binegar: The coefficients or separately zero that's the definition of linear independence so because the factory only some a linear combination of them only some to 01 only issue that makes them legally and and now recalling some definitions from last week.
Birne Binegar: We have a set of generators know them create the whole every state and they're linearly independent and what we call a basis.
Birne Binegar: The.
Birne Binegar: Rule of basically as I originally introduced is that bases provide coordinates.
Birne Binegar: Now, this was done in the contact the substation you found it to be is equal to say the one.
Birne Binegar: UK.
Birne Binegar: If that's a basis.
Birne Binegar: For subspace.
Birne Binegar: say all it's called ass then.
Birne Binegar: What we can do is, we can coordinate all the factors that live inside that someday using this basic The idea is that well the generate the entire subspace so.
Birne Binegar: Every job you sitting inside the subspace can be written actually uniquely written.
Birne Binegar: As a linear combination of these vectors So who is going to equal.
Birne Binegar: See one he won dot dot dot you see can.
Birne Binegar: You can.
Birne Binegar: When I introduced stacy's I talked a lot about organization, the idea or the purpose of the basis which provides coordinate Rebecca is that live in tigers update the idea was that you use these expansion coefficients as covariance.
Birne Binegar: See one two ck.
Birne Binegar: Are the coordinates.
Birne Binegar: Given factor inside.
Birne Binegar: are given such things.
Birne Binegar: Alright, so i'm.
Birne Binegar: Also, a basis but they're not so much space they actually generate or in itself but.
Birne Binegar: He always thought ah, so you can think of this example being a special case of this construction.
Birne Binegar: what's really special about the standard.
Birne Binegar: Is that.
Birne Binegar: we're general that for general subspace you have a basis, then you get a set of coordinates for each vector that lives inside that basis.
Birne Binegar: When you use you're talking about and your face is the entire factory and your faces.
Birne Binegar: What turns out, you should coordinate that curve is the same as that after you started off with.
Birne Binegar: Why, why is that what you can do I don't have it written down here it is.
Birne Binegar: If I say, I have a vector w.
Birne Binegar: And I represented as a linear combination my standard patient factors well when I grabbed the coefficients in this expansion and write down with use them as a list on the vector I get exactly the same vector I started on this would be.
Birne Binegar: A specter with was.
Birne Binegar: X one X am.
Birne Binegar: I expand it grabbed the coefficient to produce a coordinate vector but it's the same vector I started.
Birne Binegar: Alright, so I never thought that much about standard basis, even one well, we were I hate to use that word, because I didn't want to confuse you.
Birne Binegar: The coordinate the coordinate vector is the same as the backdrop and but that's only true for the standard things.
Birne Binegar: right but we'll use that back now.
Birne Binegar: you're saying once you expand any better terms of the standard basis according back to you get exactly the factory.
Birne Binegar: yeah oh i'm using this stuff here, because in general.
Birne Binegar: The coordinates.
Birne Binegar: Thinking about an arbitrary up space, I have many bases for this, and so my coordinate system is actually going to depend on my choice of things.
Birne Binegar: What i'm saying here is that when I use standard basis by coordinate vector.
Birne Binegar: Tonight yeah by expanding in front of my face that you're grabbing fishing that that Cohen five.
Birne Binegar: For the standard.
Birne Binegar: Alright, so.
Birne Binegar: i've got two ideas presented that I need to prove the theorem is that, given the linear transformation there's no Sophie matrix and actually replicate the action.
Birne Binegar: But then having to think of key function that we find two components of X, rather we can think of it as being simply matrix multiplication a column for fun.
Birne Binegar: Right and that this formula here tells you exactly how to construct this major what we do is T gets applied to n dimensional vectors and result after we apply the function isn't any national director oh.
Birne Binegar: For each, this is an empty space, though they'll they'll be an state your face that's where the spectrum states are yeah I applied.
Birne Binegar: function to do now that's going to be valid, because each of the standard basis after is an element of the factors like to compute these numbers, then what this.
Birne Binegar: is saying is like the beauty of numbers, I get a list of n numbers and are right, that is, a call.
Birne Binegar: But what I get yes mm columns in each calm and I rested resolve my client vertically there's gonna be in entry, so these columns and rows.
Birne Binegar: So this.
Birne Binegar: linear linear transformation, it goes from n dimensional vector space n dimensional vector to that I didn't have a true with em columns in road or without an n by n bakery.
Birne Binegar: My plan is that when I construct this matrix in this fashion, then you know transformation is action can be replicated by major my major.
Birne Binegar: is not necessarily giving you a simpler way to figure out with the Texas regional i'm use it is it.
Birne Binegar: Working with major season much closer to the calculation of said once you have a concrete matrix you just have a list of numbers are ready to recommend will re and that's because you deal with numbers here now naturally computed.
Birne Binegar: So let's prove this it's really simple.
Birne Binegar: Suppose we have this make instructed in exactly this faction.
Birne Binegar: I want to show that.
Birne Binegar: The original.
Birne Binegar: The original function is action can be replicated by matrix h.
Birne Binegar: O T applied X well i'm gonna write that, as he applied to well, I have a vector and fight for our Am I can expand it with respect to the same basic but i'm just replacing f5 trying to respect the same basis.
Birne Binegar: And then, what are the properties of T he is a linear transformation, so that means I can delay vector of issue until after I like the information in the result be the same, but keepers are that tradition.
Birne Binegar: But now, when I look at where he is acting acting on scalable.
Birne Binegar: A linear transformation also preserves scalable he so I can pull these factors X one through the action of tea, I can throw in really actually he until this line is they call it, this because he preserves patient.
Birne Binegar: Now next step is directing it to you want it's the first column with this maitre D to be the second term in this expansion.
Birne Binegar: And five right that out, and then the last step.
Birne Binegar: No life on that nature to them that matrix multiplication can be implemented by taking components of the factor you multiply and multiply it by a particular calm.
Birne Binegar: You may.
Birne Binegar: Have, this is the definition going for period here's the definition of how it got calm.
Birne Binegar: And then the last identity is a matrix multiplication anytime you have a call back divine matrix that's equivalent to taking a linear combination of a lot of the calls of the matrix in both occupy the components of the better.
Birne Binegar: Right, and so there it is to find an arbitrary vector X, is it going to be exactly the same thing as major trouble at this matrix.
Birne Binegar: All right, well let's use an example.
Birne Binegar: So i'll give you a particular function between two factors faces.
Birne Binegar: And i'll show you first of all that it's a linear transformation.
Birne Binegar: And then i'll show you that.
Birne Binegar: Will explicitly construct the matrix that corresponds to that function.
Birne Binegar: function, it takes an input free numbers and output is to number two or three two or two.
Birne Binegar: So it's a linear transformation because well what we'll do is, we want to verify that fee of scalable for X is same thing is that scale multiple on here that was verify that.
Birne Binegar: So what i'll do is i'll stuff to for X its expression in terms.
Birne Binegar: of scale location completely before I like the intervention, but i'll find the true for X one all so or use this formula, but also mandate one planet X two.
Birne Binegar: X three.
Birne Binegar: Well, as first component here is the difference of the three components of the already Oh, we have.
Birne Binegar: landed X one minus X and the second component, the sound of the.
Birne Binegar: components why sone Nice, I get the second.
Birne Binegar: Every component has a common factor Lambda I can pull that out, and then I noticed that.
Birne Binegar: This particular lecture is exactly the image of F1 do it three by this function and so now I verify that he feel about.
Birne Binegar: Anything that's.
Birne Binegar: On to that and we've shown that this function is compatible with steal patient.
Birne Binegar: If this had been.
Birne Binegar: These has a nonlinear function of the variables X one X two, this will not hold and fan of these has a linear function to the constant term this, for example, but because these are linear function about constant term, this is actually a key.
Birne Binegar: Alright, so.
Birne Binegar: that's it behave properly respect the scale addition next we'll get back tradition.
Birne Binegar: Here all by.
Birne Binegar: The function and apply that to a solid to better X, one of the components actually why I add them together now all i'm Chief of this.
Birne Binegar: Prevention factor with component X one plus y minus y.
Birne Binegar: The rule, especially in.
Birne Binegar: The first component is the difference of the first two components that will have X plus y minus X, Y Z for that person and this formula here also tells you how.
Birne Binegar: Oh yeah do is collect together the X, the terms, involve the action the term the moment one moment, and then we can break this apart into the cell to two dimensional.
Birne Binegar: The first one, and the identified as ti X.
Birne Binegar: You just do X and the second one is key to why until ricotta design again to find to an arbitrary someone factory in part three, is always going to end up being the same thing as the son of God asks us to some lot so we're compatible respect to that tradition.
Birne Binegar: showing the compatibility respect and scalable application and vector tradition makes this a linear trend for me alright, so all i've done so far and you're not going to have to do this gentle gentle way i'll tell you, you have linear transformation, you can Sue property.
Birne Binegar: For the first example I want to show you how you verify that.
Birne Binegar: If somebody is a linear transformation, you can check it out.
Birne Binegar: Alright, then now.
Birne Binegar: Now what we'll do is we'll take this function that we've now proved to the linear transformation or construct corresponding matrix.
Birne Binegar: That means we look at the action of tea on each of the standard basis that you're with a domain or three, what is the due to the first game.
Birne Binegar: It says 100, to the point, one zero just by applying the world it says second standard vector minus one lot and it sends the first danger danger factor in the vector 01.
Birne Binegar: All right now.
Birne Binegar: morning.
Birne Binegar: Great this take this emphatically now, because this is the most common mistake on the second exam.
Birne Binegar: That looks like pretty close to matrix here.
Birne Binegar: But in fact at is not going to equal.
Birne Binegar: One zero minus point 101.
Birne Binegar: That is not the right matrix and I can show you why it can't be that matrix.
Birne Binegar: Suppose when we're looking for a matrix when T such that when I do matrix multiplication of a vector and sizes domain, we get the same thing as applying the function directly now what if I use this matrix let me call this killer.
Birne Binegar: and well.
Birne Binegar: This is this night let's call this some other spurious name be.
Birne Binegar: Some other matrix but i'll show you is this matrix cannot possibly satisfy this identity, because.
Birne Binegar: He is taking.
Birne Binegar: Three dimensional vectors the two dimensional vectors.
Birne Binegar: If I write down this matrix.
Birne Binegar: 10911 and 01 and I tried to multiply a three dimensional vector by.
Birne Binegar: Well it's not defined.
Birne Binegar: Because this is a matrix 23 rows two columns and we're trying to multiply matrix three rows and one call, but these inner dimensions don't match, so the matrix multiplication is not going to work so.
Birne Binegar: Although this is the first thing to write down as you try to compute this remember that this matrix that you get by just.
Birne Binegar: Taking this manual mention of number three that's all major what you have to do it hey these guys are converting the call for you right to me.
Birne Binegar: back the matrix shot.
Birne Binegar: Several slides that I guess, this is how.
Birne Binegar: we're going to compute the guy but we're not going to write them to draw better than this column.
Birne Binegar: And what what happened here is that that would make this the right mix exactly going to be at is equal to one zero minus one one and 01 because this is in fact the matrix I can find find key to 100.
Birne Binegar: To 2010 is he on 2001 and writing results vertically.
Birne Binegar: Okay.
Birne Binegar: So.
Birne Binegar: we're we're actually working on that page here so here it is.
Birne Binegar: So when I take these results, applying to each of the standard basis factor and our right results vertically horizontally, then I get the right nature's.
Birne Binegar: river the form of this matrix.
Birne Binegar: This is it say matrix I had before, but now, when I do make multiplication of archery 3D medical badger by the matrix correctly constructed.
Birne Binegar: It produces this result and now notice that the first component is at one minute effects to it agrees with the first component of it apply to an arbitrary and the second component X two X three here agrees with the second component of the original function, so this is all working.
Birne Binegar: But it won't work I stress again it won't work unless you remember to convert the results of applying it to the standard basic factors, you have to remember to convert those the columns.
Birne Binegar: alright.
Birne Binegar: So.
Birne Binegar: We have about eight minutes left.
Birne Binegar: What we have so far.
Birne Binegar: Is a way not only to take a matrix and produce a linear transformation, but now we have a way of translation and attaching to the linear transformation matrix so, in fact, what we have is a way of going from.
Birne Binegar: linear transformations.
Birne Binegar: From.
Birne Binegar: em dimensional space.
Birne Binegar: magical place we can go back and forth between these guys and and by him matrices.
Birne Binegar: So, so, for example, if we have linear transformation.
Birne Binegar: T from.
Birne Binegar: Our em and then we can produce a matrix.
Birne Binegar: a subset of team, by applying the linear transformation that everything basic factor in writing them is called.
Birne Binegar: A call.
Birne Binegar: Or if we have a matrix.
Birne Binegar: problem that we can produce a linear transformation, by saying that he stops grip a.
Birne Binegar: vector we obtain by.
Birne Binegar: multiplying interpreting X as a column vector and then multiply by ED so from the linear transformation, we get a matrix it from a matrix we can get a linear transformation of course.
Birne Binegar: Whether we start here we start here, they just replicate itself so if you want from a linear transformation to a matrix and then you went from that matrix back to the linear transformation, you get right back where you started from really already.
Birne Binegar: And we will use that.
Birne Binegar: All right.
Birne Binegar: So device to say that there's a very bad.
Birne Binegar: Thompson and exact middle of a war between when your transformation to bankruptcy.
Birne Binegar: And what we're gonna do is because we have a tough space associate associated matrices we should expect.
Birne Binegar: given us close relationship between these function and maintain that we should have some space it's also associated with linear transformation.
Birne Binegar: However, I not that use matrix formulation of linear transformation to find a space rather i'm going to give you a functional interpretation of what that they did, and then we'll resolve the matrix each one actually computed.
Birne Binegar: Alright, so.
Birne Binegar: Here we go i'll just probably just be able to manage tell you what the range of the linear transformation just today, that will pick up with the rest of the story, on Wednesday.
Birne Binegar: So.
Birne Binegar: i'm going to.
Birne Binegar: start with a sudden linear transformation between an md.
Birne Binegar: md.
Birne Binegar: To this is going to be some matrix and, of course, with the matrix we can always create a self paced by looking at the both day all day or no space of mentors.
Birne Binegar: will actually tell you how how to find effective way to calm space in the matrix it corresponds to give them linear transformation, but i'm not going to define through its associated matrix i'm going to define that self page on a certain function properly.
Birne Binegar: So.
Birne Binegar: Let me draw a picture.
Birne Binegar: So this is general.
Birne Binegar: function theory.
Birne Binegar: If you have a function that maps a.
Birne Binegar: b.
Birne Binegar: Special subset of be that you can create by looking at all the points of be but somehow are reachable by this function, we call that the image.
Birne Binegar: me just at.
Birne Binegar: The point or the elements be inside the set capital be such that B is equal to F of a were some.
Birne Binegar: A inside the domain.
Birne Binegar: Alright, so the sun is the setup all you need is a function between to set.
Birne Binegar: Then i'm just focusing on where did the elements of a land when they're mapped over to be and that's exactly what this me set this.
Birne Binegar: What i'm going to describe is the specialization of this construction for when the function is actually between two vectors phases and the.
Birne Binegar: And the function itself the linear transformation, so when.
Birne Binegar: That corresponds to a linear transformation.
Birne Binegar: Your transformation.
Birne Binegar: is called.
Birne Binegar: The range of people.
Birne Binegar: So, more specifically, a range of linear transformation.
Birne Binegar: This linear transformation from an n dimensional space to n dimensional space that's exactly.
Birne Binegar: The point.
Birne Binegar: Why, in the targets but it's going to be are in.
Birne Binegar: There are reachable by applying this function itself element in the domain or.
Birne Binegar: y is equal to.
Birne Binegar: tea.
Birne Binegar: or.
Birne Binegar: Some.
Birne Binegar: Inside me.
Birne Binegar: Construction of the set it just using gentle function magnitude.
Birne Binegar: But what i'll show you, is it back this range the set of every race, also an element affect our end of.
Birne Binegar: The day I created a set, but now I will show that it's both and resealable.
Birne Binegar: Oh, here we go.
Birne Binegar: first thing we have to do is show that.
Birne Binegar: it's closed and feeling location.
Birne Binegar: So let's start with a point that listen.
Birne Binegar: Again, each of the linear transformation and choose a number, and I want to show that.
Birne Binegar: scalable application of this guy by landed is still inside this said.
Birne Binegar: All right, well it's done i'm going to show that this law is, why is it ready in a scalable or why it's also entirely boy.
Birne Binegar: I need to display Lambda times, why resulting from applying to some point in the domain.
Birne Binegar: Oh, here we go to Atlanta X just chooses going it's the same thing and landed on direct.
Birne Binegar: linear transformation is compatible scalable location, but he Atlanta X, Y Z Z like just landed on one, but what is this show it shows you that this number or this vector length of time, why is obtainable by applying to this particular that.
Birne Binegar: I could tell that shows you that last time, why is an element rage, because it occurs, by applying to.
Birne Binegar: A vector in it in the domain.
Birne Binegar: And certainly it's called a thunder.
Birne Binegar: Extradition because I start off with two points that live inside.
Birne Binegar: Two points of blade inside the range, then they have to come from two different factors inside of all you got five lanes, so these guys live in the race, it had to come from this domain so right then before I want to fight this guy why to keep this stuff but now.
Birne Binegar: there's some of the factors.
Birne Binegar: It corresponds to find T zone of that you hear the summer factors is all this still going to live inside the domain, so he applied to a ton of X is equal to.
Birne Binegar: This particular back here and now display.
Birne Binegar: The sum of two elements are an earache is reliable by flying key to another vector inside the domain, which shows you that the back every point.
Birne Binegar: If you take any two vectors inside the raid the necessarily they have to come from some point in that spectrum song has to come from some point in me and so.
Birne Binegar: You start off in the range and you add back together you're still in the range and so.
Birne Binegar: i've actually i've shown on the slide that range of tea if I do this fashion open sealable ation in that tradition.
Birne Binegar: And so it's a suffix.
Birne Binegar: So.
Birne Binegar: don't worry i'll show you on Wednesday, now that we've established itself state how do we find a basis for that, and instead turn out as.
Birne Binegar: The subspace exactly identical to the call space of the associated nation.
Birne Binegar: For itself takes you spend a lot of times finding bases for matrix for column space that's what idea, because the patient for the rain okay so i'll stop here and we'll pick up with this on Wednesday.