begin the work fair today probably just I think you can stay in the statute lesss class thank you for your many blessings to us we just think euler that you loved us even when we said against you learned you your face my brain amen so get to it I did post a syllabus and you know I think the quality of a surface did turn off first light stove luncheon okay I heard a better be PI suspicious be food tracking but that's you have to do so you know something what's that that the equality of the syllabus is inversely proportional to its length so I'm happy that I cut this oh it's down to four pages I usually like four five or six so this one's the least 2030 percent matters on my mouth now if I give no syllabus for a course well that's the side of academics aren't very good but that could be infinitely good that's right anything that it would be absolutely that's exactly right so here is has linear layer off who does not happen your effort who has read stuff about the year other participant okay it's okay it's calculus secondly as I envisioned it it's an extension of linear algebra now that's not the same that if you haven't any relative focus for you not at all please in fact will learn that you're out for we need as we go so what is what is in your outside a nutshell well in your on pro is the study of spaces right which have some way to add things and some way to scale it multiply things right and and then we study linear transformations which are functions from one vector space to another which preserves the multiplication right that a nutshell that you're out for Scott so for us in here I just need to define a few more terms so in linear algebra we have some safety is the vector space the vector spaces we'll look at in here are going to go over the real numbers right that means our scalars are from the reals and so like the fundamental things we want to you know fun mental concepts that we do need on occasion or the notion of linear independence which I'm going to write out just this once the SRR star go to radiation for this it's a lie right we also need the concept of spanning and it's up all right now so what is it what does it mean for linear independence basically here's the idea if you have a set of vectors s and it's a subset of some vector space V when we say that that set of vectors is linearly independent if the vectors in s are well linear the exams they'll start doing if there's no linear dependence between them right or be more precise if the only linear dependence between the vectors and s is the trivial dependence right so if an equation state that is that you know if you've got C 1 D 1 plus stead of about plus CK be K equals to 0 so that's a linear that would be a linear dependence right if this is it necessarily independent set and so here I'm supposing that V 1 for DKR an S then this must imply that coefficients are all 0 in other words the only linear dependence of a linearly independent set it's the trivial dependence that's how we how we check linear independence will bully them kind of kind of here what's a spanning set well spanning set basically was out of this we need that the stand it might take this fan of say v1 of BK what this defined to be is just it's just the set of all possible linear combinations of those vectors all right so now this time instead of forcing the constant speed zero a love would be anything and so the press is your real stand who I'm using set-builder notation should I stop and explain that to you sure go for it so I'm saying this object things like this such that this is true but that's just saying that we have these vectors B 1 through B K and we're basically waiting waiting them by these scalars these real numbers C 1 through CK and it's just all possible in your foundation right so geometrically you can think of it like what I like to say you think of it as a K dimensional plane right but this is an abstract vector space be 5007 matrices or a little wheels or whatever so but if I say it's some kind of like abstract plane you should keep by this little bit abstract and I could also not immediately say it's K dimensional right because there's no reason that there has to be that these have to be independent unless I say so right and that brings us to our next concept basis what's the basis so basis Omega is a basis for vector space V right again this is over the riddles all right I'm going to stop saying that there is a basis for the vector space V over the realistic what do we put two things we need for a beta V basis what's that one beta's linearly independent and to me right the span of a M is equal to B oh I should mention there's another way to define span that's more in terms of like pure linear algebra which doesn't involve this formula you can also define it to be the intersection of all possible subspaces which contain the vectors v1 through VA it's the minimal subspace containing those vectors but this works for us in here so this is what a basis is is this linearly independent spanning set one of the major accomplishments of men 321 is to prove that if you have the vector space and there's one basis with finitely many vectors in it an every other possible basis for that vector space has the same number of vectors so the number of vectors in the basis it's what we call dimension some property knows closing it so the number the number and beta is less than infinity and they have his basis for me than the dimension of the e is by definition the number that's beta where I hope you understand the meaning of my notation other is it okay should I formally defined in terms of eiections to the natural numbers with anything more right yeah I'm not a nothing yeah don't think carnality this is just counting and that it nothing so transpinay writing okay so let me advance fanbases and then they also think on basis of course is it allows us to take a vector space and put coordinates on it right so what's the notion of coordinate system if theta is equal to say B 1 through B M then they coordinate D Sameach of let's say x1 v1 plus plus xn BN that's just equal to what so the coefficients are x1 through xn and so the for capacitor says you take these abstract basis factors in your place that would like the unit basis vector so here's the formula so the difference between business right here is in the abstract vector space V whereas this over here is in the set of column vectors in RN this is the coordinate coordinate map corresponding to the basis data now another way to say this which I Vonda and you should be too because it allows us a lot much laziness as we go on here is that you could just say P beta of V J is just equal to e J and then extended linearly that's another major accomplishment of that you're ultra we learn that if we define a linear transformation on the basis of a vector space then it uniquely extends to a linear transformation on the entirety of the vector space which doesn't seem like much if you're in building out but it's kind of thing step back and look at it you're saying that this function is to find out a finite number of points endpoints in the vector space there's only one function goes to the entirety of the vector space and is luckier if I tell you f of X is equal to 2 for a function from the realest of the reals and then I say find the formula you'd be like I mean let's say F of 2 is equal to 3 now find the formula for F what was the problem I introduced so I said FM to it's a good point f of 2 is equal to 3 now tell me the value tell me the function you can't do it there right unless I tell you it's a linear function in which case the formula for F is immediately known just V f of X equal FX because that's the only linear function on the reals unfortunately linear functions are actually a fine functions if you have a plus B then F of 0 is not 0 so it's a linear function more and getting its weights which do you prefer I think it's bad to kind of stop that's the only linear function because that point one point defines a life of your G and this is the generalization of that carbonaria that make spaces no and then I just threw out some notation EJ what is now yeah this is the standard basis so let me up so here's the way we define an e Jenny I component of that right it's equal to product or Delta IJ you're like what's the product development develop legends it's 1 if I is equal to J its 0 if I is not equal to J so this divide basically select for example e 1 is equal to 1 0 to the 0 and e n is equal to 0 all the ways out all we happen to get one in the last five and the length of the vector of course depends on the context but these are these are the fundamental vectors in RM in physics we don't call them like X 1 K X you have whatever PF visits me that's here any questions of the petition so of course you'd like some examples right don't worry this is not going to devolve into me trying to teach all of later on for a day that would be foolish in its last words yes so I've done that I'm not going to go into like why what I'm about to say is true it's going to make some claims but these are pretty obvious claims all right Lee let me just give you a look ahead briefly what we're doing right now is trying to find a vector space a little bit later in this lecture I'm going to tell you how we define the notion of distance on the vector space once we have a notion of distance on the vector space then we can define limits right and then once we have limits we can talk about continuity and then once we settle continuity we can talk about differentiability hopefully all of those things will happen this week let's so our standard examples oh I know I was about to say that for Altuna this is just my custom you notice that these are these look these might appear to be row vectors to you but you should understand my convention is that these are sneaky-sneaky column vectors this is actually a shorthand for 10.0 this is actually a shorthand for zero that one if I wanted to actually talk about if I actually wanted to talk about honest-to-goodness row vectors I will do like you want transpose with e10 like that I use column vectors so but at the same time I don't want to put this in the middle of typing paragraphs so this is my compromise the the price that I have to pay for it is to make this discussion at the start every course so some Brown parentheses square brackets right yeah because we take basically the point is I take RN D column vectors and conceptually I think of these as points or acceptors from the origin depending on my emphasis but standard exams for us you've got the equals to RN that's a vector space right and what's the you know the basis for that the standard basis is e1 through en right so what are the codes that with the quarterback look like here we may have x1 you want to chain me to write this I I'm wasting your time oh sorry bringing this but so remember basically the essentially here the point is VJ is equal to EJ so it just I really probably should have just written Phoebe that of X is equal to X in other words the with respect to the standard basis on RN the coordinate map is just the identity map now let's see here the equals to C n what the first home URM I am so this is the set of N by 2 matrices right now what's that look like basically definitely like 1 a 1 1 a 1 n am n last one out here is a 8m one right such that a IJ are the reals so that that's you know that's going to matrix looks like right now you see what I can do is I can write this let's say per year a in B as a equal to a sum over I equals one to the sky goes from my throat it's just one am J's the column in that's in my current version your tables 110 those are not reserved I'm not saying eyes always the row index and J is on the column index that obtained I'm saying for this formula right here it's the road colleges and IJ e okay big now this e IJ is the standard matrix unit so get d IJ the kale component of that here let me just let me break it down for you real real real may secure if I have a 1 1 a 1 2 a 2 1 a 2 2 C I can write that as a 1 1 1 0 0 0 plus a 1 2 0 1 0 0 plus 81 0 1 0 let's see if you do 0 0 0 1 all right and are you just letting it if certainly getting the list so I probably should erase that well my thumbs up now okay so this is what we call a 1/1 this guy is e1 to this one if you won this one is e to - all right in fact those are linearly independent and you can see for example that I can build any 2x2 matrix by these right so in fact this set of these things gives you a basis for the set of matrices right and what the how do you define like what's the definition for this it's only zero Billy one when I is equal to K and when J is equal to L so the formula for that you can use a pair of kronecker deltas it's Kronecker Delta IJ Kronecker Delta JL see that this this is only equal to one when both of these are one this is only both is only both okay so then if you list these in terms of the lexicographic order which is I mean these are actually or this one this one this one this one right that gives you the standard coordinate map of the bankruptcies so here here it is P beta a is equal to M will be really easy here it's a 1 1 start with u 1 1 going out to a 1n so that's the first rebel right and then you do the second row all the way up to the em throw em 1 ay ok so that the coordinate map on that buying real matrices let me it's just all you do is just take the rows and just you know play about ones one at a time No so for the sake of no I can't do that yeah I'm going to have to chin to the board here a bit alright so the next try we call this G 1 & 2 all right because I'm going to talk about four examples okay go to my other two examples B 2 is C I think it's good to have some examples of complex matrices some of you would like to understand quantum mechanics right I think so I'm going to try to work some discussion of like permission spaces into the homework give you a chance to I mean it's not it's not really that sophisticated in terms of the complex analysis of it pretty much it's just basic you know you've done phasers in electrical it will be very sort of networks into that kind of algebra so I OC n is just n n n complex factors right so what that looks like is Z is something like c1 that it okay now each one of these is a complex number right so each one of those can be expanded into a real and imaginary part so that's like x1 plus PI by one right xn plus PI Y but then I can pull out the real scalars right I can rewrite this and I'm not going to write you can see if you can see what I'm writing though all right the first thing you tell me what x 1 x 1 then 10 t1 right plus y 1 x 1 xn times 1 + ym x 1/u follow links for me are you all right yeah that's I here so in fact the natural basis to use for CN is just e 1 ie 1 e to ie 2 and somewhere that's on this in fact is a linearly independent set over the reals right and that's and then so what is the corporate networks okay so let's see is see as same as it close up here you just do what you just list out real imaginary real imaginary real imaginary just like that okay so obviously the dimension of our n is an what's the dimension of the N by n matrices upper arm how many when corporates are the bigger this is an element of our web all right about dimension Evon yeah so for this one well 2 by 2 2 plus 2 is 4 2 times 2 is 4 this is a horrible thing to look at trying to figure anything out about it but yeah this is also 2 times 2 is 4 them if you had 3 by 2 matrices that's a 6 dimensional space right how about this this is an element of our 2n okay what do you think my fourth example is complex by complex yes yes so yeah oh and some of you probably were thinking that's three something to do the same example over again I'll take that see this is the FIM all right I'm going to just write the court if a phrase design P beta see let's see vmin complex matrix you know it is and so let's let's let does e equals x plus that may be tiring that X plus I Y where these X big ups and big y are both real matrices so what is the court event here it's just X 1 1 Y 1 1 X 10 Y and and if I tell you the quartet map you can also figure out what the basis was what is the basis that I didn't write the basis yet what is the basis here right okay so let me just say he I J come up ie IJ suitably ordered that as to avoid a bunch of racial anyway but of course a basis ISM basically we talk about bases in vector spaces basis and when you're talking about an ordered basis right if I switch the order of things you get a different court amount right it's kind of important right cuz that's like changing your head west or north east alights flipping the meaning of those should matter when you think about the notion portman right so we have talked whatever traces all right those are standard examples not what we can deal with those of course we can look at soft spaces of those things right so what's that was the subspace the subset from a vector space it's also a vector space a subset of vector spaces also vector space very good so just as a bit of notation in a subspace say W is a subspace of V that that's that's how we did of that right and so there's lots of cool things you know about subspaces if you have a subset a non-empty subset of a vector space which is closed under addition and scalar multiplication it's a subspace right that's the to space two-step subspace test another popular way to see that some things of subspace is is it's the span of a set of vectors with the subspace right so those those things are good remember all right so just over to that about linear transformations has everyone a great salt and stuff so I'm just going to talk to you about your confirmations and not running because if that's okay what is a linear transformation you have to Becker spaces right so I'm going to write just a script will appear a little bit so then your transformation something like T from V to W is a linear transformation it's a function for VW right such that line X plus y is equal to P of X plus P of Y right and also T of C X is equal to C times T of X that has to be true for all vectors x + y 4l go constant C so that's a Nativity and home today you have those two things you have a linear transformation rate now can you represent that by matrix multiplication yes but there's some fine print for an abstract vector space we have to run it through the course that's right that's kind of an involved thing you might not have seen it in linear algebra view 10 to 21 your profits you so I probably will have a homework for you guys that tells you what walks you through that it's not it's not super complicated or anything it's just there's something really special that happens when you have a linear transformation from column vectors to column vectors right the special case is that if if T is is a is a mapping from say RN to RM then what can you do you've got T of X is equal to standard matrix of T times X right so we put in brackets like that that's standard matrix of T but if I'm an abstract vector space I don't have necessarily a standard basis to go to I mean I told you these are the standard basis for these things but it's not universally agreed-upon okay so for pretty much anything except for RN unless you have a special context you have to state the basis and there's like a lot of fine print you have to add there is a formula for an abstract vectors that terms of matrix but again we have to run it through the port maps all of you don't work on them but this is very simple right so if we're working in RN to RM linear transformations are really really simple because you can just write them in terms of this one matrix multiplication so called standard matrix right so eventually in this course after just a couple lectures will just be an RN so then this will be true and we will have to face too much of the coordinate stuff but the start the first few lectures I want to try to stay up with the level of what's called a normed space so that you appreciate the generality and also we can do some really neat problems about funny life setting fabulous of matrices and things which i think is interesting let's hear I'm not planning to give you guys another homework problem so it's important to like refresher agree on these things that really is not hard the other thing I'm going to give you a little bit about is the direct sum decomposition write a vector space V is the director 7e composition of p1 plus p2 this this plays an important role as you study engine spaces in normal spaces as we go on here so I'd like to explain that to you a little bit and so this just means that everything in the is either in v1 or v2 but it's it's it's also uniquely so so like the only thing that's in both we want to be 2 is 0 if you give me a vector X and V there's a there's an X 1 and X 2 X 1 D 1 X 2 then V 2 so that you can get back to it I mean this is really it goes people like the y axis and x axis but ok I'm not getting off track now so listen I um I know it's a lot to ask of you to review all in linear algebra that mean my notes are like 300 pages that's in kind of me so what I've done because ever like a little review chapter that is it's pretty dense but in about eight pages or so I flush out this and a lot of details and so if you read that you should be at least good up you know you should have some moral appreciation for layer up for us we won't actually use all those things so I'm just trying to make you work some of the issues this is usually the case of my course if you can do the homework you're urethane you can't do the homework you asking for help okay so I hate to Riesling sales by detective that brings us to the study of normed spaces oh by the way I think this is about the most theoretical art in the course currently I just tried to review a three-hour credit course in space of 15 minutes or whatever is that maybe seems like a bit much especially you haven't got it do not display or if you have just had it for me do not just say I'm mostly trying to make you aware of language that we're going to use norm spaces and I'm actually not here's the definition so we're going to talk about the vector space alright so be a vector space over the reals parent with we use this notation like double vertical bars that's the norm more so be it's real value okay and it satisfies a couple of important characteristics right so here's the definition one two three four now intuitively speaking what we're trying to do is we're trying to extract the notion of vector length so the properties of this should be like essential features of the length of a vector that you do study said calculus three right so let's see here first of all we know that the length of a vector is a start one so what if we do scalar multiplication after that one I multiply a vector by two it doubles the length right if I multiply the vector by -2 it's still not least the length just in terms of stretch around right so the port 11f this is culpable unity is just you can pull out the absolute value of the scalar function right so there's family homogeneity or some people say absolute imaginating you've got this again the ordering some people might put this class not know yeah this one this is called the triangle a quality if you think about x and y being the horizontal and vertical next to a triangle so if you got X to get Y and then X plus 5 is here so of course the length of the hypotenuse is less than the sum of the lengths of the legs and then also this what we see in terms of inequality this is what greater than or equal to 0 and sometimes people will separate so many people will combine very important to one thing not just making it two things so that I don't lose anybody a diamond sorry it's not negative but when is it zero what is the norm zero right if and only if or I could do the double arrow biconditional right the filly has equal zero these are the necessary ingredients to call the dysfunction of normal in a vector space with a norm is called the norms that you know vector space or if we want to be a little Russian in our lingo we could say that all of this together you know be paired with let's say V say W be here with the norm is a normed linear space so we'll call it an n LS normed linear space I'm sorry what is the meaning of the double bars the double bar is you know so I'm getting the set of axioms if you have some function on a vector space that satisfies these four things it's an order this this notation is just a shorthand for like I mean the actual function and it's just a notation so what are the norms and okay so once we have the storm we can do other things like let me just get to it what's the definition of an open ball with respect to the normal so yours I can talk about open ball open ball means markers so for example an open ball radius epsilon centered at X is 1 let's say let's say VND such a well for some reason I want our ad minus X but I can eat your you know the norm of X equal to the norm of minus X call us from the absolute homogeneity it's equal minus one you get but yeah so this is so with what Daniel is doing is he's what's this thing he's doing here like we we understand if you've studied physics you recognize that it the displacement vector right so what you're really saying in the distance between V and X is given by the length of the displacement vector from the point X to the point B that's where this is coming from but if this has to be less than funny that's that's not right so this would define essentially an open ball in the North linear space and we actually just stumbled upon another thing which is the distance function what's the distance function the distance function D from let's say DX is equal to the norm of t minus X so you can define a distance function once you have a normal if I can just say from the clouds here for a minute more if you have a set paired with a distance function that's called a metric space what is a distance function what makes distance distance again another set of axioms okay so like distance from X to Y is equal to the distance from Y to X right same time to go it's not like your commute to work the distance let's see here the distance from X to X is 1 0 right the distance from X to about 80 left repeat myself the distance from X to Y is 1 also it's greater than or equal to 0 right so these 2 things that I just wrote collectively referred to as positive definite metric is positive definite and then you also have this distance from x to y plus the distance from Y to Z is greater than or equal to the distance from X to Z this is also called the trying to link walking that's the triangle inequality for norms this is the triangle inequality for a metric now if you have D it's a mapping from subspace cross itself back into the reals right if you have this packing D if it satisfies these axioms that makes em into what's called a metric space I said nothing about linear algebra here you can you can give them that you can make all kinds of weirdo shapes into metric spaces there's no notion of vector addition necessarily in this so this is a much much fuzzier notion and that's not really the profits of this class this is something you'd study properly in analysis okay that 431 should study this carefully should study sharply it has so if it's an or linear space it's automatically a metric space because you can induce the metric by just this vector length idea you cannot reverse this though because this does not assume the structure of a linear space it's not about the n-dimensional vector space so you can traverse this this is this idea there's another important thing which you have been exposed to how do we did have had we when I teach physics you guys are physics me writing how do we know how we defined the vector length what do we use to do it the one yeah but the what I use to do it the dr. crane remember we're just very immersed in the dot product where's the dot product all that's what's the what's a generalization of that product well that that's what's called an inner product space so yet another yet another abstract notion which I should share with you because it's just a decent thing to do is put a verification is that there's something called an inner product space now an inner product space which I will put four more over here so if you have the pair foot and so sorry more of this kind of fuzzy notation Bradley right yes I love you're listening it's amazing oh so this is a mapping from the cross feed into the reals if these are real vector space okay so I'm talking about a real inner product space it's a mapping and it has to have the following things the inner product of say X plus y the Z is equal to X comma Z plus y comma Z and likewise X with y plus Z is equal to X comma y plus X comma 0 cos it's infant you can also pull scalars out of those basically it's a linear transformation in both slots it's got this linear linear linear because five constant here you can pull it out I have a constant here so it's so physics with a file in your mouth linear both lots and what else makes the dot product about products you know yeah how about a dot B sequel to be tied right so an inner product has to have at symmetry and it also has to have that the inner product of X with itself is greater than or equal to 0 and since I'm separating it the inner product of X with X is equal to 0 it W what X is equal to 0 ok so these things open is there a triangle equality probably so so this is these these defines what it means to have the envy a vector space over the reals paired with such a bilinear rail this makes V and inner product space see these three competing notions of distance inter product nor this in metric space right they're related but they're not the same right can you see how to go from here to here from this world of that world plate to go from here to here it's easy what I do is I define the norm of X equal to the square root of the inner product of X and X right that's like that's the formula like the length of a is 8 the square root of a diet but done in the language of inner products it turns out that you sometimes reverse the zero it's actually possible to take a norm take a norm and build an inner product sometimes you can do that but I was this using something called the parallelogram law so I would actually give you guys a homework problem where you can flush that out a little bit it shouldn't be hard early mostly yourself okay so again basically this arrow just goes that way but definitely the reverse arrow can't happen now there's a lot of things that you can you know products are really important to geometry especially in RN like are in a product for RN is gonna be the dot product so basically that captures our notion of things being perpendicular and so one of the big stories in this course is going to be you know the study of tangent spaces in normal spaces and so that's all about the dot product but all right I got that just wanted to throw those terms out for you now how would be how would we define that this is an open ball with listen this is evolved right it I don't think it's an open ball what does it mean to have an open set what's an open set looks like so I'm going to I mean you raise everything here except to the north space for any focus again dark spaces now I just I wanted to give you that little feel about the connection between and a product spaces to stand and FM the metric motion whatever space D tensor I think it's important to have that big picture even if we're not studying metric spaces this semester so much okay how do we find limit I mean a more basic question though is what's it open set right so what I'm trying to notice right for you it is oh I'm sorry there's a very I'm being very rude I haven't given you any examples of workspaces so we should do that so v1 is a normed space right we just it's just the usual thing right I'm referring to the v1 that I erase so the norm of X is just square root of x is right it's a square root of X 1 square plus X 2 square plus xn squared right so RN is a normed linear space to prospective usual notion of length of a vector v2 v2 was why it was our M by n so what's what do you think the norm of a matrix should be this is the city um but I'm going to be very lazy my suggestion to up to you guys is what you can do is you can just say it's the length of the corresponding coordinate vector so I would say that it's just the length of the beta of a where that was this was we just took a string down Row 1 Row 2 all the way up to row M and here I'm referring again to this notion just usual the usual Euclidean length basically just take the matrix string it out as if they think long and by n vector and calculate the length of that in our Emma that's the that's the length of matrix weird right what's the length I think you see which is going with the length of a complex vector I think you don't know right you're right it's just the length of the corresponding to M to n dimensional real vector with where you string out the real imaginary parts what's the length of what's the length of a complex matrix this is starting to get boring right it's just the length of the corresponding real coordinate vector now I'm proposing that we you can use those as our go-to definition all right actually this is by no means the only notion of length you can get through these sets in fact there's a lot of freedom for a given finite dimensional vector space there's actually infinitely many different the party the trophy norm is hard from being pink all right okay well you just said we're going to find limits in terms of the norm right that seems a little bit disconcerting you told me there's definitely many different notions of norm does that mean that the limit is going to be different if I use different norms that would be annoying right thank you know the limit is actually insensitive to the choice of norm because all the different norms you can look at actually generate what's called the same topology what's it what's a technology oh that I'm digressing there's this is kind of this is not terribly as if I mean that this is kind of like obvious now but something really magical happens for these examples in that there's another formula which is which is slick for these this is actually also called Li the for being this one so the perviness norm if I put a square it's easier it's the trace of a transpose a that's really really awesome because I am relating the proposed length or proposed geometry of matrices to this specific algebraic operation on matrix that's very precious and also this formula that kind of thing in an arbitrary vector space you don't have some notion of multiplication and you have no hope of such a formula what's the formula for complex vector there's also something there you just take basically take your vector and there's different ways to write this but here's one way to write it I'm square just so I have square roots I think if I take B and I multiply it by whilst I say B dagger times B Duffee dagger is what it is is it is V one kind of in the end conjugate so basically it's the conjugate transpose this is also known as the provision adjoint it's important operation in quantum mechanics okay so this that basically just gives you the sum of the modulus modulus squared for person Connaughton second one and the third component it's like x1 squared plus x2 squared x1 squared plus y1 squared is X u squared is y 2x + that plus xn squared plus yn squared or we have those switches it is it is just the sum of the squares of the real and imaginary components it's just a slip formula for that but that's a nice make copulatory and so with this in this down here what you have very similarly the formula for the do the trace of Z dagger Z and that actually is a formula for the normal so if we can only work in more with these things that would be kind of possible right because once you use this coordinate that you're pretty much during away everything you do about the matrix as being a matrix because this court Ahmet just lists things out you lose track with like what matrix whole thing what does matrix multiplication look like the global coordinates something awful so okay getting back then to let them how do we define limit I have about 45 great thank you there's two ways I can go at this point lecture I could either tell you more about how we define limits or I can show you more about different norms for a specific space I'm going to go ahead and define a limit all right and then I'm going to go back and give you a little bit more why the choice is norm is not unique in today's 12:30 not reportable to chance at 12:45 show what about in the next classes oh there are people uses building truth - that's true but you know that can tell me something else ok so what does it need for a set to be open come so here's a picture what is it open trap basically this is sort of so this is you what it means for you to be open in the metric sense right so and when I say metric sense I mean as judged by this as a metric right so also known as the network topology the metric topology with respect to me so it's obviously with respect to a choice of form the thinking of this is about space but if you want if this is too abstract for you think about it being r2 doesn't mean for a you know like to set mark do to do but it means that if you've been a point in their life at you can find a little open disc that is around that point P and it's inside the set that makes P what's called an interior point so U is open so let's say you subset of B it's open if for each e in you there exists epsilon greater than zero such that the ball of radius epsilon is entered at P is contained in that makes P an interior point if every point in the set is an interior point it's an open set now where's Annie percent to be closed it's a closed right that's our definition definition a negative s s is closed if the minus s is open and the B epsilon of P is just the absent equivalent of my drawing a circle around T right it's it's it's a epsilon ball in the normal linear space oh yeah peanut so that's - sphere matrices do it we have the necessary definition you will look for higher spheres of matrices we're not going to do it but your imagination is all kinds of things that's open that's closed other things we need to kind of I'm just well on the topic what's the boundary point where each open ball has points in the interior and the exterior yeah right if you did put the point is out of the boundary if when you look at that point and you take any open ball around that there's a thing every open ball around another point contains points in and outside the sentence now the boundary point itself is not yet to be in the set right so but for every choice for every radius there's points inside that's the topological boundary anyway topology don't be scared of it is just a word that means basically study of continuity and hand-in-hand the study of what's an open set they do a lot of personal ology here at some point but all right so get back to now what's our definition of limit and what we mean by if I say the limit this one's I've got a function from one more linear space V to another one W and I want to say the limit as X approaches a path of X is equal to B over there over that what should that mean bunch of that mutation