48 - Linear maps

we're ready to start our next big topic of this course which is linear maps they're also called linear transformations same thing and the first thing I want to do is write the definition then we're gonna discuss it a bit and understand it then we're gonna list some examples and then proceed with the theory so here's first the definition so far everything we did we always worked in the context of a specific vector space the vector space was sometimes just RN and tuples thought of as row vectors or column vectors sometimes the space was a space of matrices sometimes a space of polynomials but always a single vector space all the guys were taking from a single vector space okay and what may be interesting and is interesting whenever you have a notion in mathematics this is really a general approach in mathematics whenever you have a notion for example vector spaces the next thing you would be interested in is functions of those of that notion so what are functions between vector spaces okay functions from one vector space to another vector space okay and a function in general you know that from calculus a function in general is just the rule that tells you to every element in the in the domain what the element and the range to it what is the element in the range to which it it Maps right a unique element in the range okay but whenever we're talking of maps or functions or transformations which relate to specific to specific structures we want those maps to preserve the structure okay so the idea is a map between two vector space a map from v1 to v2 we would want it for example if we add two guys in v1 a plus B and then transform we would want to get the same result as if we first transform and then add in v2 if we take a vector here and multiply it by five and then send it we want to get the same result as if we send it and then multiply it by five okay we want the map to respect the structure the underlying structure the underlying structure in terms of vector spaces is not much there's addition and there's scalar multiplication we want the functions to respect addition and scalar multiplication that's it that's a linear map so let's write that let V 1 and V 2 B 2 vector spaces they do have one thing in common in order for everything to compile they have to be over the same field they have to to obey the same set of scalars okay over some field F the same F for both okay a function what a function is you know a function but we're going to usually denote our functions with tt4 transformations rather than f like in calculus t from v1 to v2 a function T from v1 to v2 is called a linear function or a linear map or a linear transformation if it satisfies two things one if you take two little guys in v1 two vectors U and V add them and then apply the function you get the same thing as apply the function to one apply the function to the other and then add T of U plus T of U that's the first rule okay note that on the board right now I have this little cross here in this little cross here there are both plus signs but they're very different this is the addition in v1 here I'm adding two guys from v1 this is the addition in v2 I'm adding T of U and T of V which are elements in v2 do you see that I'm writing the same plus but in fact those are two different operations they live in two separate worlds good the second property is respecting scalar multiplication if you take T of alpha V so alpha V is a scalar product of a vector in V 1 and then duty to it send it to V 2 you get the same thing as Duty 2 V itself and then multiply that by alpha this is scalar multiplication in V 2 and since it's the same set of scalars alpha is the same alpha in both that works so this is the definition of a linear map I should just add here that this is true for any U V in V 1 and for any Alpha in F good ok so the way to think of a linear as just a function a function is just the rule sending guys from here to guys from there any rule okay that satisfies these two properties which means that it respects it conserves the vector space structure the linear space structure of the underlying V 1 and V 2 okay good let's do examples so here are some examples example 1 so in order to give you an example I need to tell you what V 1 is I need to tell you what V 2 is and I need to tell you what the map is what T is and then verify that it indeed satisfies those two properties ok so here's the first example V 1 is gonna be our 3 of X do you remember what this notation stands for right polynomials of degree less than or equal to 3 over R with coefficients in are good so that's gonna be V 1 and V 2 is going to be R 2 of X polynomials of degree less than or equal to 2 ok and we're gonna find define aa map a function from V 1 to V 2 so in order to define it I have to tell you what it does to a given polynomial of degree less than or equal to 3 so let's take a polynomial of degree less than or equal to 3 let's call it ready p4 polynomial and what does T do to P of X it simply takes P prime the derivative everybody remember what the derivative is even if you're not there yet in calculus right good everybody okay so do you agree that the derivative a part of a polynomial is always of the degree of a lesser degree than the original polynomial if is gonna be an x squared up two coefficients right so do you agree that it this indeed goes from polynomials of degree less than or equal to three two polynomials of degree less than or equal to two okay now the question is is it linear so T is linear linear meaning it satisfies the two properties that we had in the definition Y so the first property was take two polynomials in r3 call them P of X and Q of X add them and then duty to the Sun what does Duty mean Duty means just take the derivative of whatever you have P of X plus Q of X prime right now we know from calculus that the derivative of a sum is the sum of the derivatives right so this equals P prime of X plus Q prime of X this is true not just for polynomials but for general functions right and this equals what is P prime P prime is T of P and Q prime is T of Q so this is T of P of X plus T of Q of X so T of a sum is the sum of the T's that's the first property based on previous knowledge that the derivative is in fact a linear map the derivative of a sum is the sum of the derivatives good here's the second property what if what is T of some scalar times a polynomial it's the derivative of the scalar times the polynomial that's what he does takes the derivative right again we know from our previous knowledge of the operation of taking the derivative in general for functions that if you take three times a function and take the derivative it's the same as three times the derivative right so this equals alpha times P prime of X right and this is precisely alpha times T of P of X so the Alpha hopped out of the T so these two statements are satisfied and therefore T is a linear map the derivative acting on polynomials of degree three sending them two polynomials of degree two sorry less than or equal to theta less than or equal to two is a linear map it's not just a function it's a linear function good okay by the way we're refraining from calling T a function where although it is it's a function between vector spaces but we're calling it a map or a transformation precisely for this reason Z could be a space of functions so this is a space of functions right polynomials are functions T acts on functions the input and the output are themselves function functions do you see that okay it's like one level up the spaces are spaces of functions and T is a function between functions okay so that's why at least terminology wise in order to keep things more less confusing we're calling T not a function but a map good okay here sure yeah so you're asking can we define different teas let me write some examples we can do many many teas okay we can define various kinds of linear maps we just have to check that it that they're linear if you take a function and divided by 2 that's a map okay that taking a function and dividing it by 2 is a map is it linear you have to verify ok you have to verify these two properties sometimes it's straightforward the verification sometimes you have to work under stand the question no no this is the domain and the range of T a function is comprised of three things domain range and rule this is the domain this is the range and this is the rule you have to give all three ingredients in order to specify a specific map or function but we're not calling it a function okay domain range and rule okay good here's another example here's another example example number two here T is gonna go from R in to R M so these are just n tuples over the fields are but but these are n tuples and these are M tuples vectors with n entries and vectors with M entries and it's gonna be defined as follows I have to tell you what he does again this is the domain this is the range now I have to tell you what is the rule right so the rule is tell me what T does to a vector V is gonna be a vector in RN I somehow have to produce a vector in RN okay so here it what it here is what T does it multiplies V by a specific matrix a where a a is a specific a given M by n matrix so if you give me a vector suppose M is 3 and n is 2 if you give me a and and I fix in a 3 by 2 matrix okay if you give me a may a vector V I multiply it by this matrix a and I get a vector of length 3 if if if n was 2 I get a vector of length 3 okay so I want to argue that this is a linear transformation in general and then I'm gonna do an example of the example so I'm gonna take a specific a and show how it works okay so this is a linear map T is linear how do I show that it's linear so first abstractly okay fix a why is T linear because I have to check the two properties what is T of U plus V T of U plus V is by definition what does T do it multiplies U plus V by this matrix 8 right what do you get well we know these are all matrices right a is M by n these are vectors are n by 1 right so the multiplication is well-defined right and we get an M by 1 vectors but we know rules for matrix multiplication this is the same as a u plus a V right we know that make matrix multiplication and addition satisfy this distributivity but what is au au is precisely T of U and viii is precisely TFE do you agree and likewise again abstractly take a vector multiply it by a scalar what does this specific T do it takes a and multiplies it by this vector and again we know that the scalar can move freely among matrix multiplication so this is the same as alpha times a V right and this is alpha times T of e good so multiplying by a matrix is a linear transformation okay this is an example this is example 2 and in fact there's going to be two sub a which is going to be a concrete example where I'm gonna write a concrete matrix here but in fact in fact this is not just an example this is something completely general in the following sense any linear transformation between two vector space can be encoded in terms of a matrix can be translated to multiplying by a matrix okay so this is something we're gonna discuss it's gonna have some details and we're gonna have to build up towards it but this is a spoiler any linear transformation can be written in this form okay between any two vector spaces we're just gonna have to translate from the original vector spaces to the coefficient vectors and then T is gonna be encoded by a matrix t is gonna have a matrix representation okay spoiler this will show up later but just keep it in mind this is very general what we're doing here okay okay so let's do a concrete example you can agree that there are millions of examples here even without knowing what I just said take any matrix you get a different example okay so let's call this example to prime an example of the example so T is gonna go from r2 to r3 and the way T is gonna work on a vector in T is gonna operate on a vector in r2 a vectors in r2 is something of the form XY it's gonna do it's gonna take the matrix 1 to 0 negative 1 and multiply it by X Y sorry this is not our three this is art two again sorry everybody fix this okay so T goes from r2 to r2 the way T works is multiplying the vector in r2 by this specific matrix a this is a in this example okay so this is very concrete okay so let's verify that this is indeed a linear transformation although we did it abstractly let's do it more concretely so we can feel it so we can see it with our own eyes so what is T so let's verify that T is indeed a linear map so we need to check two things namely the two properties of being a linear map here's the first one we need to take T okay tell me if you agree to what I'm doing I have to see what it does to X y plus what did I call it Z W that's T of a sum of two guys do you agree okay so what is this well now it's very concrete it's this so it's one two zero negative one this is a times this factor I know how to add in our two right this is just the vector X plus Z y plus W do you agree okay and now I know how to do this multiplication what do I get I get X plus Z plus two times y plus W X plus Z plus two times y plus W right and the second component of the vector is going to be zero times X plus si minus so minus y plus W this is what I get do you agree okay what happens if I do t to the first one plus T to the second one let's see what we get there T of X y plus T of Z W we better get the same thing right let's see if we do so what is T of X Y it's 1 2 0 negative 1 times X Y plus 1 to 0 negative 1 times Z W do you agree good so what do we get we get X plus 2y x plus 2y and - why that's this plus Z plus 2w and minus W do you agree and obviously we know how to add vectors and this is the same as this it's the same vector good so these two are equal therefore T of the sum is the same as the sum of the t's that's the first property of being a linear map good ok so this is equal to this good the second property of being a linear map let's take T of Alpha X Y by the way did you note that I made a slight abuse of notation what's called okay so I should have written T of another set of parentheses X Y right this should have been written this should have been written as T of the vector X Y right but usually we we don't write it like this and we just write this and this is called abusive notation it should be this but this is so much clearer that we just do it clear was that be more clear okay but when we when we have the Alpha inside we have to put the other pair of parenthesis so we get this so what is T of Alpha X Y it's taking the matrix-12 0 negative 1 that's what T does multiply by the matrix and multiply what this vector this is just a vector alpha X alpha Y right that's this vector good and this equals let's see what we get we get alpha X plus 2 alpha Y that's the first component and then 0 times alpha X minus alpha Y do you agree we could either multiply separately to find what is alpha T of X Y but here in fact it's very easy to just pull the Alpha out right so we can pull alpha out and we're left with X plus 2y and negative Y right and this we recognize that's this this was precisely T of X Y right so this equals T of X Y that's this thing multiplied by L good okay so we just verified manually although the the abstract general statement said it very clearly okay we verified manually that this specific specific matrix here on specific and an M indeed gives a linear transformation okay so I know that at this point it's important to do this really with your hands to get the feeling but I think it's also a good opportunity to make a certain remark okay you can see the power of abstractness here look how we worked how we kind of wrote down all the details in order to verify that this very specific linear map is indeed a linear map whereas and look on this board it's just one example of a very general thing that looks very elegant to write down like this and includes any au right here for any RN and any RM or FN and FM the the fact that it was ARB didn't play any role and you can see the power of abstractness right you understand what I'm saying the point I'm trying to make this is it abstractness has its its merits let's see okay you have to remember that when you start with something abstract it may be too abstract may be confusing and you may need to strip it further into a very concrete example in order to really understand what's going on but once you do understand that it's it's a very powerful tool good okay so I want to we're gonna see many many many more examples but gradually because I want to gradually introduce more terminology and notions and then each example is yet another one and I'm gonna exhibit one more notion or term or so but I do want to right now I want to write a very small theorem very basic fear and very easy theorem and prove it and a certain remark which we're gonna discuss at length but not right now gradually so first of all a remark so let's say this suppose you take T of a vector X Y and define T of X Y to be let's say X 1 this is a perfectly legitimate function right going from R 2 to R 2 do you agree so this is a map T going from R 2 to R 2 for example okay but this map is not linear okay this is not it's a map okay it's a function it's well defined but this is not a linear and you can try checking those two properties and see and you'll see it right away which fail okay so this is not a linear map and it raises the question when I write something of this form what do the components in the range have to satisfy in order for it to be a linear map okay suppose I write here for example X 0 would it be a linear map suppose I write x squared Y would it be a linear map and so on okay and the answer the answer we're gonna have is that the components each have to be your combination of these linear combinations of these components in order for it to be a linear map okay so you can't have squares there you can't have just standalone scalars there but zero is good zero is a linear combination of x and y right x squared is not 1 is not good but we're gonna see that later on but you should practice this and try writing several things and seeing manually again to get the feeling that that is very important when you when you're introduced to something new to really touch it to touch it okay so write down several things like this and verify if they're linear maps or not until you get the feeling okay ah okay and here's the theorem that I wanted to mention a very very basic one if T from V 1 to V 2 is a linear map so if it does satisfy those two properties of being a linear map then two things hold the first one is that T of 0 is always 0 and the second is that T of the additive inverse minus V is always minus T of V ok let's prove this very short and immediate from the definition so what we know we know by the definition of a linear map let's take a peek at the definition again a linear map is just a map from V 1 to V 2 look at this board please a map from V 1 to V 2 that satisfies two properties and the only property we're gonna need in this proof is property number two that when you multiply by a scalar and duty it's the same as doing T and then multiplying by the scalar this is the proper we're gonna use okay so back here we know by the definition by definition that T of alpha V equals alpha T of e for any alpha in any V right so this is true for any alpha in F and for any V in V one write the domain space right in particular in particular right if we take alpha to be zero it should still be true because it's true for any alpha by definition so if we take alpha to be 0 we get T of 0 times V T of 0 times V equals 0 times T of V do you agree but 0 times V is what the zero vector so this is just T of 0 and 0 times anything is 0 so T of 0 is 0 and that proves a do you agree and if alpha equals negative 1 another legitimate scalar in F right the inverse of the unit every field has has a 1 in it right what do we get we get T of negative V negative 1 times V which is just negative V in fact we even proved that that negative 1 times V is minus V I think we proved it at the very beginning when we started discussing vector spaces remember that so T of minus V is minus one times TV good so that completes this proof okay so these are not part of the definition but they're as close as can be to being part of the definition and we're going to use these facts freely in what's to come okay that any linear transformation always carries zero to zero that's a very easy easy way for example of touching and seeing if map is a linear transformation or not here's an example is T of zero zero zero zero no T of zero zero is zero one that's it it's not a linear transformation doesn't satisfy this do you see good so this is a very tangible way of determining if a linear to a very easy way of determining right away of for linear transformation is linear or not okay it's not an if and only if statement but it's a way of stating that a linear that a map is not leaning okay okay so we still this is just the introduction to what a linear transformation is what we want to do next is define some more terms some more notions and then kind of write more examples and in them include already the new notions and discover what they are so that's coming up next

Channel: Technion

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Keywords: Technion, Algebra 1M, Dr. Aviv Censor, International school of engineering

Id: -M9QWpJa-ls

Length: 37min 19sec (2239 seconds)

Published: Thu Nov 26 2015