[Music] hello and welcome to the online bfc program on data science and programming in this video we are going to continue from our previous video where we introduced vector spaces and study some properties of vector spaces in particular we will build on the axioms that we have seen in the definition of a vector space so let us derive a consequence of the axioms the conditions that are needed in order to for something to be a vector space so this is called the cancellation law of vector addition so if v one plus v you have three vectors v one v two v three in the vector space v and v one plus v three is v two plus v three then v one is v two ok now ah let's first see how we do this in rn so let's try to recall how we do this in rn so in rn what do we do so in rn what we do is let me do it for example in ah m three r three z one z two z three plus y one y two y three so how will i conclude that x one x two x three is the same as z one z two z three well what i will say is this means x one plus y one is ah z one plus y one or more generally x i plus y i is z i plus y i where i is between one and three i is one two three and then from real numbers i know that x i plus y i is z i plus y i implies x i is equal to z how do i know that because if you [Music] add minus y i on both sides then indeed what you get is x i z so you can subtract y i from both sides you know subtract y i from both sides okay so this will imply x i is z and from here we conclude that uh two x three is the same as z one z two z three okay so this is how we do it for uh real numbers sorry for r three or in general for any rn ah so then let's see how to do it i mean we can use the same idea to do it for general vector spaces so e one plus v three is v two plus v three so this was the argument in rn ok so now we are given ah that v one plus v three is v two plus v three this argument for general vector spaces because v three is v two plus v three well one of the axioms told us that um for v three there there exists um some v three prime such that ah so there exists v three prime such that v three plus v three prime is zero so what that tells me is v one plus v three plus v three prime is v two plus v three plus v three prime but now we know that addition is associative so i can write this as v 1 plus v 3 plus v 3 prime is v 2 plus v 3 plus v three prime um so which implies that v one plus zero is v two plus zero which implies v one is v two yeah so this proves a statement here so notice that all i used was the axioms yeah i did not use any other thing any other information about the structure of v because really i do not know anything else yeah the only things i know are the axioms ok so whenever you want to check something for a vector space you have to use the axioms but to get intuition about whether it is correct or wrong and how to go about a proof in case it is correct you should look at rn see how you do it in rn and then ah sort of extrapolate that idea for general vector spaces ok so here is a corollary of the cancellation law the vector 0 described in 3 is unique so remember one of the axioms was that there is a vector called 0 but maybe there are many vectors satisfying that same property so this corollary says you cannot have many vectors and here is another corollary namely that if you looked at the vector v prime this was in axiom four which so this axiom four is what we used in order to conclude the cancellation law by looking at v three prime right so the vector v prime described in four is unique and it is standard to refer to it as minus v yeah and ah even this reference to minus v will be clarified in a upcoming slide okay so maybe let us quickly do this proof from the cancellation law so suppose i have another vector called w such that w also satisfies the same property that v does so suppose there exists w in v such that v plus w is v for all v in v right this was what ah zero satisfied ok but then v plus w is v means what v plus w is v plus 0 right because v plus 0 is also v yeah so i can write it like this but then i can cancel yeah so i am adding v on both sides so i can cancel v so the idea is here you add v prime right and so that leaves us with ah w 0 so cancellation means i can do this that leaves us with w 0 okay so the vector 0 is disk as described in the third axiom is unique okay and similarly the vector v prime described in 4 is unique and so we have v plus v prime is zero so now suppose v double prime also satisfies this so then v plus v prime zero is v plus v double prime ok and now you can use cancellation so cancel v ah and therefore v prime is equal to v double prime ok so that is that is the proof of this corollary from the cancellation law fine lets ah look for some other properties so in any vector space v the following statements hold true ah zero times v zero for each v ah minus c times v is minus of c v and that is the same as c times minus v so what is minus v v minus v is exactly the vector v prime ah which was described in axiom 4 and our previous slide told us that it is unique and hence we can refer to it as minus v because we know that that is a vector so that when you add it to v you get 0 okay so c times 0 is 0 for each constant in our e scalar c in r so let's try and prove this so what i can do is i can look at zero plus zero times v and one of the axioms tells me that this is zero times v plus zero times e on the other hand zero times v ah is sorry zero plus zero is exactly zero which is zero times v right so what is the net result and that result is that zero times v zero times v plus zero times v so you can write this as zero times v plus zero is zero times v plus zero times v cancel zero times v and that implies zero is equal to zero times v so that gives us the result ok and then we have minus c times v is minus ah c v is c times minus v so how do i get this the idea is exactly what we have done above so we have c plus minus c times v ah is c times v plus minus c times v but on the other hand i know that this is zero times v which we have just proved zero so c times v plus ah minus c times v is zero yeah so that means minus c times v ah satisfies the condition required of the negative of cv okay so this implies minus c times v is minus of cv and then similarly you can check for c times minus v ok ah finally we have c times 0 0 and maybe i leave this to you because it is in a very similar spirit to what we have done earlier so check this fine so lets do an example sort of from so to say from real life ah this is an example that we have seen in the previous video ah the example of stock taking ah so what do we have here we have stock taking in the grocery shop so we have five items rice and kg dal in kg oil litters biscuits and packets and soap bars okay and the first column describes how many of which quantity is in stock the second column describes the demand for each product for buy array the second third column describes the same thing for buyer b the fourth one for buyer c and the last one describes what is the new stock that has arrived so what is i mean we did vectors in this context by looking at the corresponding column vectors and and so on so but we can also think of this as a abstract vector space so what is the vector space showing up here [Music] so the vector space is the quantity of rice in kg the quantity of dal in kg the quantity of oil liters the number of biscuits or the number of biscuit packets and finally the number of soap bars yeah so it is this set okay and so the claim is that this is a vector space why because if you add you add coordinate wise and what you get is so if you have two two vectors in this are added you get the total quantity of rice within those two vectors the total quantity of dal in kg the total quantity of oil liters the number of biscuit total number of biscuit packets and the number of soap bars total number of soap bars right that is how you do addition and scalar multiplication is ah coordinate wise ok so it so happens that this vector space looks very much like r five ok and why is that because we have five quantities ok and of course here now once we write it like this we have to also ask what does one mean by negative quantities and negative number of soap bars so this again we saw in that example ah what negative corresponded to so negative corresponding to demand and positive corresponded to supply so negative corresponds right so for example if you have minus 2 kgs of rice that means that there is a demand for 2 kgs of rice right and if we say that there are 3.5 liters of oil that means there is a supply of 3.5 liters of oil um and then we can use this plus and minus in order to take stock this was exactly how we did it in that example ok so um of course one has to also interpret things like what is half a biscuit packet or what is one eighth of a soap bar so one of the things that we do as a result is that we often express them in units for which we can have any real number instead of only natural numbers right so um then it seems more natural so for example if instead of biscuit packets we had written 100 grams of biscuits if we know that let's say one biscuit packet is 100 grams okay or if instead of soap bar we had written 400 grams of soup yeah then there is no issue with decimal places or what we add and subtract ah so this is another example of a vector space and maybe we will end with this example of a fine slats which you are going to see again later ah so suppose we have so this is a slightly geometric view of vector spaces ah and a deliberately convoluted one so that i want you to ah sort of also be comfortable with the geometry involved here so yeah maybe before i start the example i should point out that much of what we will do is formal algebra but behind the scenes there is geometry which is guiding our algebra right whenever we say r3 or r two ah in our mind we we have geometry yeah we understand ah two dimensional space or three dimensional space ok so suppose v is a plane parallel to the x y plane so we are going to define an addition and a scalar multiplication of points on v so scalar multiplication is going to be done as follows so let q belong to this plane and let c be a constant so we project q under the x y plane scale the resulting vector by c and project the result back to v yeah this is the procedure for doing scalar multiplication so what is c times q c times q is the tip of the obtained arrow ok this is the procedure ah for scalar multiplication ah maybe let me draw a quick picture here so so here is your plane here is your point so drop ah this perpendicular to the x y plane and then draw this line here and then if you want to scale this by c so if you scale it by c then that means we are going to get some new vector something like this and then project this back up to the um original space so you will get may be something here ok so this is your point q and this is your point c times q okay that is scalar multiplication ok so here is how you do addition [Music] ok and we are going to watch this edition uh this is thanks to our support team ah so here's your plane which is parallel to the xy plane and now we take two points on this plane which we want to add so now we draw our usual vectors ok and then we project this down ok and now so now this is in the xy plane so in the xy plane we know how to act because that is exactly r2 that is the parallelogram law so you use the parallelogram law add them and then you project the entire thing back to your original plane that you are working with and now the newly obtained point ah which you can see over there is the sum okay this is the way you add so i hope the video showed you the geometry involved in things like addition ah so now the question is so we have defined order scalar multiplication and addition on this set and now i leave you to check that this is a vector space ah you can either do it geometrically or better to do it both ways so try to think of geometrically by visualization why this is a vector space or you can just write down the algebra and and work out that this is indeed a vector space so in either case what can help you is the following so the idea behind the addition and multiplication scalar multiplication here is that really you are looking at vectors we we are taking points drawing the corresponding vector but then remember that you project down right so this is essentially what we are doing is we are looking at arrows which start at the point where the z axis intersects this plane and where the tip is the point you are interested in okay so you take those kinds of arrows and then you do addition and scalar multiplication for those arrows exactly the way you do it for r2 this is this is really at the heart of what is going on ok so i will leave that visualization to you and i hope you can see that this is indeed very similar to what we do in r2 okay so let me summarize what we have seen in this video so in this video we saw we began by seeing the uh some x some more properties of vector spaces in particular things like how the 0 vector behaves and how the negative of an of a vector behaves and so on we saw somewhat real life application of vector spaces namely we took our good old grocery shop problem and saw how vector spaces fit into that context and then we we have seen the example of affine flats thank you you