Birne Binegar: today.
Birne Binegar: Is do I just posted homework Sep seven homework that six dealt with mining bases for the sub spaces that are attached to a matrix.
Birne Binegar: So we did lots of examples last week but i'm wondering if there any questions on those web assign problems.
Birne Binegar: Well, hopefully that's good news, so what we're going to do today is I introduce last time, especially some spaces or special subsets that are.
Birne Binegar: The most important subsets of a vector space that one considers in linear algebra and similarly for functions.
Birne Binegar: We have a special class of functions that are relevant than linear algebra, these are the so called linear transformation, so I introduce those.
Birne Binegar: Last Friday we're going to talk more about them today, but the main thing I want to do is take the abstract stuff all the definitions and stuff and related to our concrete.
Birne Binegar: calculation of setting of matrices So the first half of today's lecture it's going to be to take the idea of a function between two vector spaces and associate with it a particular matrix.
Birne Binegar: And then we'll find it or there's a really tight association connection between matrices and linear transformation basically it's one to one.
Birne Binegar: For every linear transformation, you have a matrix and for every matrix you have a linear transformation.
Birne Binegar: So that should suggest, particularly in light of what we did last week that since mater seeds have like three sub spaces, that we can naturally associated with them the span of the columns span of the road and.
Birne Binegar: The solution set the corresponding linear system well because linear translations are related to make her sees.
Birne Binegar: You would expect, with their some special subspace is associated with a linear transformation, just like there are four major cities, and in fact we're going to.
Birne Binegar: Talk about two sub spaces associated with reading your transformation what's nice about these is their definitions, even though they relate directly to the sort of Sub spaces, we considered for matrices they're defined.
Birne Binegar: Just using functional language, rather than linear algebra language, yet when we try to calculate them will see that we go back to matrix nice to actually figure out precisely what.
Birne Binegar: A given range of a transformation is what the Colonel linear transformation is.
Birne Binegar: Okay, so let's um I mean yes.
Birne Binegar: Start today's lecture we're doing.
Birne Binegar: Real quickly what linear transformation is we've been looking at these vector spaces throughout the whole semester now we're going to consider functions that map one deck your stage to another.
Birne Binegar: What we require for linear transformation it's it should take us from one vector space to another.
Birne Binegar: But it's also got to behave well respect to our fundamental vector space operations, so if we scale or multiply a vector before we apply the function, we should get the same result as if.
Birne Binegar: We just applied it directly to be and then postpone the scale of multiplication till after we applied to function.
Birne Binegar: and, similarly, we should be able to add vectors before we apply the transformation or after we apply the transformation and we would get the same result.
Birne Binegar: So it's it's basically a compatibility property that we're imposing on functions between our two vectors faces we want them to be compatible with our two fundamental vector space operations.
Birne Binegar: Okay, so that's.
Birne Binegar: that's those are the object oriented dealing special functions, but which these properties can be maintained.
Birne Binegar: Now one.
Birne Binegar: Really prototypical example of how you make a linear transformation is to start with any matrix in rows and columns once you.
Birne Binegar: Have a matrix specified, you can create a linear transformation, it takes you from well number of columns you have that corresponding rm to the number of rows you have our end and the way that's done is you.
Birne Binegar: take it back you're inside the domain you reinterpreted as a column vector so that you can perform matrix mold vacation so remember how much matrix multiplication works.
Birne Binegar: If you have.
Birne Binegar: An n by n matrix.
Birne Binegar: And you have a column vector so matrix with em rows and one column then what's going to produce from this matrix multiplication it's going to produce a and.
Birne Binegar: All you have to match, you need, and then the outside dimensions give you the dimensions of the product, so you get an end by one make matrix which you can think of.
Birne Binegar: As an ordered list of and numbers, and so this guy lives inside our am but after we multiply by a we get an element in our end.
Birne Binegar: So that's easy easy way to create a function between two vectors spaces and.
Birne Binegar: What we what we do to find the linear transformation that will add to this matrix is will simply define it to action on a m dimensional vector as being matrix multiplication by you're given matrix all right, and then it's easy to see that this is a linear transformation because.
Birne Binegar: let's see t of Lambda X it, let me write this yeah P Lambda X, let me put her subscript a because.
Birne Binegar: i'm going to find this function in terms of this matrix as well that's all the subscript needs on specified a now i've got a linear transformation and what this is going to be I the definition of this function is just going to be a times well the argument is Lambda X.
Birne Binegar: But scale or multiplication commutes with vector matrix multiplication So this is the same thing if a.
Birne Binegar: We can just reverse the order of those modifications, and then we get Lambda times a actual use Lambda times T subscript apex because a times X is exactly what we mean by this function T subscript a.
Birne Binegar: So here, you see the compatibility with scale or multiplication and, similarly, if we have some of vectors.
Birne Binegar: And we applied this function to it well, I definition of this function, this is a times X plus y, then I can distribute that.
Birne Binegar: Over the vector some.
Birne Binegar: And we get to have X plus t of why, and so we also have the natural compatibility with vector edition so both scale multiplication and vector addition are preserved by this function.
Birne Binegar: What i'm going to show today is that the converse is also true.
Birne Binegar: This talent, this is saying that if I have a matrix I can create a linear transformation, the goal for today is to show that you have a linear transformation, you can find a corresponding matrix.
Birne Binegar: Solid state that's a little more precisely anytime you have a function between two vectors spaces that are which.
Birne Binegar: scale or multiplication and vector addition are preserved.
Birne Binegar: Then there's going to be a corresponding matrix i'll label it a subscript T rather than T subscript gay, because now, the input is a function and our output is going to be a particular matrix.
Birne Binegar: And my claim is that once I have a function that has the properties that I can find a matrix such that the action of tea on any vector can be replicated by.
Birne Binegar: matrix multiplication this is sort of similar to what we did with elementary matrix each what we had elementary row operations and we found that we can implement them.
Birne Binegar: directly by matrix multiplication now what i'm going to show us that you have an arbitrary ocean well, not an arbitrary function but an arbitrary linear transformation between two vector spaces, that that can be implemented by matrix multiplication.
Birne Binegar: Okay, but.
Birne Binegar: The show this it's going to require a couple simple.
Birne Binegar: Short digressions.
Birne Binegar: When will need a better formula for matrix multiplication in particular.
Birne Binegar: Alright, so that is in fact the content of that first lemma how to.
Birne Binegar: find a simple statement about what it means to multiply of column vector by a matrix so the statement was quickly show is that if you let me just denote by column subscript J a for the jade column of the matrix.
Birne Binegar: Meeting.
Birne Binegar: This column J hey would be the column vector let's see i'm going to be in the jade column, that the roadmap index is changing a to J a nj So these are the entries of the jays calm, with a general n by n matrix a equals.
Birne Binegar: A one one a one m a n one and M and all the entries in between.
Birne Binegar: So that's that's all this noting i'm just grabbing a particular column of this matrix and i'm denoting it by.
Birne Binegar: column J, so the statement i'm going to make is that, when you multiply a column vector times a matrix the result.
Birne Binegar: can be thought of as a particular linear combinations of the columns of a and in fact the coefficients.
Birne Binegar: In this expansion are exactly the entries of the vector this being multiplied by a.
Birne Binegar: So if you know the components of X and you know the columns of your matrix then no alternative way to compute this product is just take this linear combination of the columns all right, so why is that true well.
Birne Binegar: It really can be demonstrated just a couple lines right here's a matrix product an arbitrary invite him matrix times, an arbitrary n dimensional column vector.
Birne Binegar: We compete this by taking the dot product, the Rosa the first factor with the one and only calm and the second factor and we get this.
Birne Binegar: But now we can do an operation that we we've done many times, for example, when we solve linear systems will expand this particular vector refresh back to these parameters X one X two up the exam.
Birne Binegar: X one is multiplying first call of a you look what excellent it will playing in each term and each component i'm getting.
Birne Binegar: An entry in the first column times the first entry of.
Birne Binegar: X and then the second column you look at it it's just going to be.
Birne Binegar: The entries in the second column of a but they're all multiplying the second component of X two So if I simply expand.
Birne Binegar: The vector that results from doing the matrix multiplication if I expand it in terms of the variables X one X m you see that I get exactly a linear combination of the columns of the original matrix.
Birne Binegar: Okay, so we'll use this identity gives us another way alternative way of computing the matrix product.
Birne Binegar: All right now the second thing I want to.
Birne Binegar: talk a little bit about is the so called standard basis for our m we haven't used it that much, but let me remind you it's just.
Birne Binegar: The easy basis you get by.
Birne Binegar: Take taking all n dimensional vectors and specializing to the ones were only one component is non zero and that component is equal to one.
Birne Binegar: So the first standard basis vectors where the first component is one and all other components are equal to zero, the second standard basis vectors we're only the second component is non zero and the second component is equal one etc.
Birne Binegar: But we have these so called standard basis directors for our em.
Birne Binegar: But now you see the purpose of our language here.
Birne Binegar: If I think of these justice an arbitrary set of vectors sitting inside our em then.
Birne Binegar: Well, one thing is obvious, is, I can get it, I can reach any vector insight on Am I taking linear combinations of these guys.
Birne Binegar: right because i'm.
Birne Binegar: pretty easy to see well, in fact, the next slide will show you that.
Birne Binegar: Well, not so easily okay so.
Birne Binegar: What i'm saying here is that you take a vector X one X m, then you can think of it as X one times one plus x two times two plus x m times E m.
Birne Binegar: So, because.
Birne Binegar: You just use this definition for the standard basis vectors this identity is sort of odd, yes, but what it shows is that every vector inside.
Birne Binegar: Your n dimensional space can be expanded and as an written as a linear combination of these standard basis vectors you want to eat yeah.
Birne Binegar: So that's the first point that every director inside your vector space is a linear combination of these guys, but that doesn't mean that these guys are basis, yet you not only have to be able to reach every point in your space, but you have these.
Birne Binegar: You need these guys to be literally independent if you're going to have a basis but it's pretty clear that these guys are linearly independent or maybe the simplest way to demonstrate that is to notice that.
Birne Binegar: A linear combination of these vectors is going to equal to zero vectors.
Birne Binegar: If that equals to zero vector that each of these components has to equal zero, and that means that each of these expansion coefficient separately happy equals zero so.
Birne Binegar: X.
Birne Binegar: equals zero this linear linear combination equals zero only if.
Birne Binegar: yeah 00 equals.
Birne Binegar: X one X one plus dot dot dot plus x and Pam if, and only if.
Birne Binegar: X one equals zero X is equal to zero.
Birne Binegar: And X is equal to zero, so the only linear combination that produces the zero vector is a trivial linear combination and that's the definition of linear independence so.
Birne Binegar: These particular vectors.
Birne Binegar: Not only generate the entire vector space there linearly independent, and for that reason they're a basis but that's just recalling our definitions of bases and linear independence etc okay so.
Birne Binegar: One thing to note about this basis and.
Birne Binegar: The reason why I never talked about it.
Birne Binegar: I may have mentioned it, but I haven't really dealt with it, that much.
Birne Binegar: If you recall, I introduced bases as a way of coordinate vector spaces, the idea is is that if he equals the one through bk.
Birne Binegar: Is a basis.
Birne Binegar: For say a subspace.
Birne Binegar: Then.
Birne Binegar: Yes, can be coordinate ties.
Birne Binegar: What do you do you take a vector say little s inside so.
Birne Binegar: The fact that you have a basis for that these factors means that s has an expansion, in fact, a unique expansion in terms of these basis factors.
Birne Binegar: si K.
K.
Birne Binegar: So.
Birne Binegar: Once you have the basis that means every vector can be expanded.
Birne Binegar: In terms of your given basis and, moreover, these coefficients are unique and so what we glean from that is that we can.
Birne Binegar: ascribe coordinates to this director, you say see one two ck.
Birne Binegar: Are.
Birne Binegar: The coordinates.
Birne Binegar: Given vector s inside a subspace and that was.
Birne Binegar: that's what we needed, because in general.
Birne Binegar: The individual vectors are going to live in a large dimensional space, but the subspace we're interested is a blower dimension, so we want to have our coordinates attuned to our subspace not the space in which the vectors actually live well what happens when we do that.
Birne Binegar: That construction to the standard basis factors well here now, the reason why I never talked about it because it turns out to be really trivial, in this case.
Birne Binegar: You have a.
Birne Binegar: An arbitrary list of em numbers here it's written now, then you can expand it in terms of the standard basis vectors because these, in fact, are a basis for our am.
Birne Binegar: But then, according to this rule when we went to coordinate ties rm what we could do is expand in terms of the basis and then using the coefficients as the coordinates so when we do that when we grab.
Birne Binegar: The coefficients and expansion, with respect to the basis and we put them in a vector well the vector we get it's the same as the vector we started off with, and so the standard basis vectors unique in that.
Birne Binegar: it's coordinate vectors anytime you expand a vector in terms of it and grab the coefficient you get the same vector that you started off with.
Birne Binegar: Okay, so that's just peculiar to the standard basis sector what we've been I never mentioned this before, because.
Birne Binegar: Actually, our goal is, I was trying to teach you about the basis is not to rely on the standard basis factor.
Birne Binegar: Okay, so.
Birne Binegar: All right, so now i'll show you the connection between a linear transformation and how I can get a matrix from a function between two vector spaces what we'll do is i'll for a matrix i'll express all constructed explicitly and here it is.
Birne Binegar: So each of these standard basis vectors or.
Birne Binegar: R m it's good in particular it's going to be an n dimensional vector so this function can be applied to it and i'm going to get an n dimensional vector as a result.
Birne Binegar: What i'm gonna do is i'm going to take the result of these and i'm going to write the resultant vector vertically horizontally or vertically.
Birne Binegar: And i'll do it once for each of the standard basis factor so with an m dimensional there'll be an standard basic vectors, to which I can apply the linear transformation in there by generate a particular column of this matrix.
Birne Binegar: Alright, so the claim, then it said, if you construct this matrix.
Birne Binegar: Using the standard basis vectors for the domain and the function itself, then in fact matrix multiplication by this matrix replicates the action of the function when applied directly to the same vector so I can either operate directly on the.
Birne Binegar: elements of our aim by applying the function, or I can implement this function, by doing a matrix multiplication.
Birne Binegar: Alright, so let me show you why that's true.
Birne Binegar: here's the definition of what this matrix is all you need to do to write it down is a specific function key and knowledge of how it acts on the standard basis sectors.
So.
Birne Binegar: So this is what we want to show and so let's start computing.
Birne Binegar: P of X it's going to be T applied to well X all expanded in terms of the standard basis.
Birne Binegar: Well that's just expanding an arbitrary vector inside the domain, with respect to the standard basis, he wants to the end.
Birne Binegar: But then, because.
Birne Binegar: linear transformations are compatible with vector addition I can delay the sector sounds until after I apply the linear transformation.
Birne Binegar: Then you move this up a little bit.
Birne Binegar: Then linear transformation are also compatible with a scale or multiplication I can do the scale of medication before I applied to function.
Birne Binegar: or after I apply the function and the result is to be the same if T is a linear transformation, so I get this and that's just because T preserve scale multiplication.
Birne Binegar: But now I recognize this guy as being the first column of this matrix second entry here will be the second column this matrix and the last summon which can be xm times the last column of that matrix.
Birne Binegar: So this expansion is exactly this and now, if you recall our matrix multiplication identity.
Birne Binegar: product of a matrix times the vector now that can be computed by taking the components of this sector and multiplying the respective columns of this matrix and then summing everything up, and so this particular linear combination of the columns of this matrix is exactly a times X.
Birne Binegar: That matrix times X and so we've got it.
Birne Binegar: The goal is to find a given a function between two vectors stations, the goal was to find a matrix that would implement that function directly via matrix multiplication and what i've shown you is that this particular matrix works.
Birne Binegar: And in fact.
Birne Binegar: If I wanted, if not now, I have a matrix if you find a linear transformation, I took this matrix and I use it to define the linear transformation via this rule.
Birne Binegar: And stack so i'm working backwards now suppose I was given this matrix and I use it to find the linear transformation well I get back to the original transformation don't back.
Birne Binegar: there's a one to one correspondence between functions that preserve scale or multiplication and vector addition and matrices.
Birne Binegar: Alright, so let's do an explicit example.
Birne Binegar: position is a little better okay so here's a function it's going to take three dimensional vectors.
Birne Binegar: The two dimensional vectors the components of the not a major just going to be the difference between the first two components and the second component of the image vector is going to be the sum of the last component of the input vector.
Birne Binegar: Alright, so first thing i'll do is i'll show you that this is indeed a linear transformation.
Birne Binegar: let's check.
Birne Binegar: let's first check that it's compatible with scaling application so i'll apply it to scale multiple X and the question I want to.
Birne Binegar: check is does this equal to Lambda times to X that's that's the compatibility with scale or multiplication.
Birne Binegar: property that linear transformation must have.
Birne Binegar: don't start computing when I do the scale of multiplication I just bring the factor of Lambda inside and now I can apply this rule for the function T.
Birne Binegar: It just on what i'm going to have to do is substitute for X one here or sub to Lambda X one X two or sub to Lambda X two into the right hand side of that that mapping.
Birne Binegar: This is what we get the difference of the first two components of the argument, and then the second component is some of the last two components of the argument.
Birne Binegar: And we have a common factor of Lambda Lambda in each term and each component, so I pull that factor out and then you see this is just Lambda times, he applied to the original vector X one X two X three, and so I verified this identity that.
Birne Binegar: function T apply to Lambda times X is the same thing same result as Lambda scale or multiply the value of T at the point X.
Birne Binegar: Alright, so T is compatible with.
Birne Binegar: scale multiplication and.
Birne Binegar: needs example i'm referring to this particular function is there a question.
Birne Binegar: Alright, so let me show you that this particular function is also compatible with that tradition, so all apply the function to have some effect years.
Birne Binegar: you're too generic factors all add them together and now i've got to apply to a particular vector of computing again, I take the difference of the first two components to get.
Birne Binegar: The first component of the image point and then i'll sum, the last two components to get the.
Birne Binegar: The second component of the image point and then i'll just rearrange terms a little bit of get the X terms group them together and the y terms of group them together.
Birne Binegar: And then, what you see is.
Birne Binegar: You can separate this vector some into terms that involve only the components of act in terms that involve only the components of the vector why.
Birne Binegar: And that's exactly he of X times, he might different of the first two components and the some of the second two components alright, so this function.
Birne Binegar: defined by this formula does satisfy the essential properties of the linear transformation that's compatible with scale multiplication as shown on the last slide and it's compatible with vector edition is shown on this slide, and so it is a linear transformation.
Birne Binegar: All right, well that's just verifying it we're dealing with a function that has the right properties.
Birne Binegar: let's now construct the corresponding matrix a subgroup T.
Birne Binegar: So the way that's done.
Birne Binegar: Again, is by.
Birne Binegar: attaching a matrix to that linear transformation, where.
Birne Binegar: The ice column.
Birne Binegar: are made, called the jade column.
Birne Binegar: Of.
Birne Binegar: The matrix a subscript T is T apply to the J standard basis factor.
Birne Binegar: don't what i'm doing in these three lines years i'm computing how this function X on the standard basis factors.
Birne Binegar: Again, a justice, a little bit so.
Birne Binegar: When T is applied to the first standard basis factor all substitute 100 for the coordinates X one X two X three and what's going to happen is i'll get one minute zero and zero Plus zero.
Birne Binegar: For the result applying T to this particular vector and, similarly, for the other two standard basis factors all right now most common mistake on the second exam i'll tell you right right away, is saying that this means that.
Birne Binegar: don't want you writing this down, so this is not me that a T is equal to one zero minus 1101.
Birne Binegar: that's not true.
Birne Binegar: Because there's one thing that you're missing here you found out how he acts on the standard basis factors but you've written result in the wrong way.
Birne Binegar: You go back to the matrix I wrote down.
Birne Binegar: Here way back several slides back yeah here it is it's just not showing up very well.
Birne Binegar: The instruction for formulas matrix is to write these guys as columns not as rows and so.
Birne Binegar: What I really need to do no sorry one way.
Birne Binegar: That.
Birne Binegar: Once i've computed the action of my given function on the standard basis factors what I need to do to get the right metrics is the right thing results, not as rose but it's called.
Birne Binegar: And so, this first row i've written here actually this first result for standard basis factor that gives you the first column the course nine matrix.
Birne Binegar: The action of key on the second standard basis factory gives you the second column that's really necessary so that that matrix will do what we want.
Birne Binegar: We have.
Birne Binegar: T that going from n dimensional vectors to n dimensional vectors and we want this to be replicated.
Birne Binegar: By.
Birne Binegar: A vector getting sent.
Birne Binegar: To.
Birne Binegar: matrix multiplication at the vector by this particular matrix.
Birne Binegar: What are we going to need Well, first of all.
Birne Binegar: Are you input vector it's going to be n dimensional so that's going to be an m by one matrix and our.
Birne Binegar: what's what is this matrix going to be it's going to be.
Birne Binegar: Well way we've constructed it it's got exactly m columns and rows so this would be an n by n matrix.
Birne Binegar: And that works, because the inner dimensions match, and so what will get out of this matrix multiplication is going to be in in like one matrix and that we can we regard as an ordered list of and numbers.
Birne Binegar: But.
Birne Binegar: If.
Birne Binegar: If I tried to use this matrix instead of this matrix it wouldn't have worked, because my input vector is still going to be three dimensional So if I took this, let me write this kilda.
Birne Binegar: it's not actually the matrix you want, but a tilda.
Birne Binegar: T times X one X two.
Birne Binegar: X three all that's going to be.
Birne Binegar: One zero minus 1101 times X one X two X three and then notice that this matrix multiplication is not defined, here we have.
Birne Binegar: A matrix with two columns multiplying a column vector with three entries, so the number of columns of the first actor does not equal, the number rose to the effect second factor, so this is undefined.
Birne Binegar: So when you look for the matrix attached to a linear transformation well given a formula like this, this is a natural way to to write the answer.
Birne Binegar: of what happens when you apply it to a standard basic structure, you have to remember that these factors here have to be converted the columns in order to get the right matrix.
Birne Binegar: Alright, so.
Birne Binegar: I don't want to belabor that point too much, because what I fear.
Birne Binegar: Is you only remember this calculation and you'll forget to convert the things in the columns don't do that.
Birne Binegar: But.
Birne Binegar: So long as you construct this matrix exactly as I prescribed here you'll get the right matrix and it will replicate the action of your function on vectors you'll replicate them simply by doing matrix multiplication.
Birne Binegar: Well, I think.
Birne Binegar: I have a slide that demonstrate that explicitly.
Birne Binegar: Maybe don't well yeah here it is so this is the matrix I constructed using the columns of tea applied to the standard using as columns.
Birne Binegar: The result applying are given transformation to the standard basis vectors and then, when I multiply an arbitrary three dimensional vector by that matrix.
Birne Binegar: I get this result and then notice that i'm replicating the action here this first component of the result.
Birne Binegar: is just excellent mine effects to as in with the original function, and the second component of the result is exactly second component of the original thing so indeed we are replicating the action of that function via matrix multiplication by this particular matrix.
Birne Binegar: OK, so now we can bounce back and forth between linear transformations and matrices.
Birne Binegar: What i'm going to talk about next is substances that are attached to a linear transformation.
Birne Binegar: So let me reread you first what we did with matrices.
Birne Binegar: Once we have a matrix.
Birne Binegar: i'll see yeah i'm just saying the connection, but once we have a matrix we consider three associated substances.
Birne Binegar: We can take the role vectors of the matrix you consider all possible linear combinations that generates the subspace of our.
Birne Binegar: Our am called the row space or we can take the column vectors a take all possible linear combinations of them that generates a sub status or in.
Birne Binegar: And last thing we could do, which we can use the matrix to write to create a homogeneous linear system and then study the solution set of the linear system that is also a subspace we call that the no space of the matrix.
Birne Binegar: Since I write down a matrix, I have three associated substations.
Birne Binegar: What i'll show you now is that you thought of the column space of a matrix and then all states of a matrix are going to be relevant also towards the sub spaces, we can attach to a linear transformation.
Birne Binegar: And it's act.
Birne Binegar: Like to do that, indirectly, I could take a matrix a linear transformation T create its corresponding matrix a subscription fee and then look at.
Birne Binegar: The rose face column space and no spaces, that that matrix a subscript T but i'm going to introduce to sub spaces attached to a linear transformation, not by referring back to the matrix reformulation of the problem, but rather by stressing the some functional properties of this function.
Birne Binegar: So.
Birne Binegar: What i'm going to look at our two special subsets one of the domain of the function here we have P, going from.
Birne Binegar: Our in our in this is your domain.
Birne Binegar: And it's a vector space and this the set the once you land on it's called the Co domain.
Birne Binegar: Now anytime you have a function.
Birne Binegar: That maps one set to another set.
Birne Binegar: You have some special subsets associated with the domain and the co domain, the image.
Birne Binegar: of F well what this consists of is all the points in the target space The co domain, be that can be reached by applying this function, so this is going to be the set of points in your co domain, such that.
Birne Binegar: That point.
Birne Binegar: is the result of applying your function.
Birne Binegar: yeah to some point.
Birne Binegar: Inside your domain.
Birne Binegar: That that set of points is called the image of your function and.
Birne Binegar: We have a similar idea we just specialized our functions to be linear transformations so that the domain and co domain are actually vector spaces.
Birne Binegar: In this case, the range.
Birne Binegar: Of a linear transformation, this is defined.
Birne Binegar: As its image.
Birne Binegar: But.
Birne Binegar: we're living your transformation, we use this more specialized language is exactly the image according to just just being a function between two sets the image is always created.
Birne Binegar: But i'll write it out explicitly it's going to consist of vectors so same vectors in the domain I sorry it's going to consist of.
Birne Binegar: vectors living in the co domain.
Birne Binegar: Such that.
Birne Binegar: That vector is reachable by applying the linear transformation.
Birne Binegar: X or.
Birne Binegar: Some X inside the domain.
Birne Binegar: Alright, so that's.
Birne Binegar: that's a set every element and that set is a vector sitting inside aren't em what i'll show you that in fact this set is actually a subspace of rem.
Birne Binegar: i'm sorry I didn't do so hard to see.
Birne Binegar: it's going to become a parent on the slides anyway, I get my.
There we go.
Birne Binegar: Alright, so.
Birne Binegar: here's what I said was the image you just take.
Birne Binegar: You take you can create the image by simply applying the function F to every point that lives in the domain and collecting everything together into a single set.
Birne Binegar: Or you can think of it as all the points in the co domain that are reachable by applying that function F to some element.
Birne Binegar: And then here's the specialization of this idea to a linear transformation, where the restriction is it now i'm looking at particular functions that are compatible of scale multiplication and vector edition, and that map one vector space into another.
Birne Binegar: And we call that the range.
Birne Binegar: So it's actually a subspace because well when we need to show we need to show that this set is closed under both scale or multiplication and vector edition.
Birne Binegar: But it's pretty easy to do that.
Birne Binegar: If I take any vector inside the range, so he is going to map some vector inside the code in the domain on to this point and let Lambda be any point in any real number but i'll show you is the scale of multiple.
Birne Binegar: Why by Lambda is still in that same set.
Birne Binegar: So.
Birne Binegar: here's the argument.
Birne Binegar: Let me start off with a point that sits inside that set inside the range and let Lambda be any real number, what I want to show is that Lambda times, why is still inside this set.
Birne Binegar: Okay now because why in the range, that means that why had to come from some point of X I the application of the functionality.
Birne Binegar: And so.
Birne Binegar: That means that t of Lambda X because it's a linear transformation.
Birne Binegar: I can delay the scale of multiplication until after I play the function so i'm move this back your Lambda through the linear transformation, but now I have Lambda times to why.
Birne Binegar: So what does that.
Birne Binegar: To get the statement I want, I have to read this backwards.
Birne Binegar: This particular vector Lambda times, why is in the range of tea because it's realizable by applying the function T to this particular point in the domain.
Birne Binegar: So the show that this guy is in the range, I have to show you that it came from some point in in the domain, but this you look at extreme side it's showing you exactly that that this point actually came from this point inside the domain and so i'm.
Birne Binegar: Set is closed understand the multiplication because if you start off in that set you multiply by a number Lambda you're still inside that set.
Birne Binegar: yeah.
Birne Binegar: So Lambda so if we will want to show a subspace.
Birne Binegar: And this is the confusing part because both linear transformations the functions and the sub spaces to set behave a certain way with for seek up to scale a multiplication or sub spaces, what we want is that if you take a number.
Birne Binegar: And you take an element that lives inside your subspace.
Birne Binegar: What you want is that Lambda times that vector is still inside your subspace and.
Birne Binegar: For you also require they fell off with two vectors.
Birne Binegar: That start off in your subspace and then you add them together.
Birne Binegar: That you're still inside that subspace, so this is requirements for subspace.
Birne Binegar: And it's requirements that we're putting on a set not on a function for linear transformation.
Birne Binegar: We have two requirements as well, but this time it's the requirement is, if you start off with a number.
Birne Binegar: And a vector.
Birne Binegar: US they call this tea from our em.
Birne Binegar: far and we start off with a number and a vector in the domain.
Birne Binegar: Then necessarily he Lambda times X has to equal Lambda times t of X so it's a compatibility property of this of this function not.
Birne Binegar: Here we're talking about the ability to stay inside the same subspace and here we're basically talking about a compatibility property, we have the ability to delay scale or multiplication until after we apply the function so they're different but they're similar.
Birne Binegar: And we also have what we have two vectors X one X two that start off.
Birne Binegar: In the domain of our function.
Birne Binegar: After we add them together.
Birne Binegar: well.
Birne Binegar: We can add them together and then apply the function, or we can apply the function to the individual vectors.
Birne Binegar: And do the vector some after we apply the function.
Birne Binegar: If these two properties hold, we have a special family of functions it's these two properties hold, we have a special subset.
Birne Binegar: So they're very similar.
Birne Binegar: And in fact i'm keeping with a basic theme of this course is that compatibility with scaling multiplication in vector addition give all the structure to linear algebra that.
Birne Binegar: In that sense.
Birne Binegar: We should have some property of functions it's analogous to this these properties subset of because they're different objects, are going to be.
Birne Binegar: Different senses of compatibility.
Birne Binegar: Alright, so what I showed yeah.
Birne Binegar: um well so.
Birne Binegar: yeah be careful.
Birne Binegar: keep things distinct I guess what I want to say is yes, we can attach matrices to these things, and in fact when we actually try to calculate.
Birne Binegar: Things like this we're going to use matrices because those are just concrete numerical objects that we can manipulate using just arithmetic so yeah there will be a eventually there'll be an implication and, but in fact what i'll show you is the sub spaces that.
Birne Binegar: We actually figure out appropriate sub spaces, that will be able to relate them to the sub spaces, that we could calculate from matrices so just bear with oh shoot i'm out of time.
Birne Binegar: Let me, let me just quickly forward because it sort of answers your question.
Birne Binegar: way, we can actually write down so i'll show this again on Wednesday, that the range of tea is actually a subspace of the domain, but.
Birne Binegar: What i'll show you is that this range.
Birne Binegar: Which is the fine fine just using this as functional definition to create the range all I do is figure out every point I can reach by applying the linear transformation.
Birne Binegar: separate from that is my ability.
Birne Binegar: Once given the linear transformation to write down the corresponding matrix what will show you next time is this range let's define just using the functional properties of the linear transformation that could actually be calculated.
Birne Binegar: As the so called space in this matrix and so, since we know how to find bases for column spaces, will also be able to find bases for the range of the corresponding matrices.
Birne Binegar: And there's another subspace that will be able to attach to effectively the no space of the matrix and that's going to be the Colonel the linear transformation, but that that discussion of delay until Wednesday let you guys go today, let me remind you.
Birne Binegar: That problem set six is do today unofficially, by the end of the Semester visually and do have look at those problems because.
Birne Binegar: we're going to be doing calculations to Ken or depend on finding bases for the non column space of a matrix bases sees for the no space for matrix so you need to be able to do that and that's so that what you'll understand what's going on, if we start to work with linear transformations.