noc20-ma54-lec21 - Vector Spaces

[Music] [Music] all right so what we had seen was we have been able to understand how to solve a system of liquid linear equation how does the matrix multiplication helps in doing things but what is more important is that there are lot of equivalent conditions which were playing some role or the other when we had square matrices fine so you would like to go further on those ideas and then understand things that is one idea the other thing is that solving a system of equation also led us to the study of vectors in some sense in the sense that if i have a solution then there was some combination of vectors which gave me the right hand side if i writing at a x equal to b then b was sum of certain vectors fine so you would like to again go on those directions so let us understand what are vectors so the main idea for us will be now to understand what are vectors all right so let us write what the vectors are so what we understand in r 2 or r 3 r 2 or r 3 that you have a vector x you have another vector y then you can add them x plus y is also a vector is also a vector fine alpha times x is also a vector so we are assumed that x and y are vectors and therefore all these two happens not only that we have that x plus y was equal to y plus x we had x plus zero vector is same as the z as the vector x for every x we had minus x such that this gave us zero then there was a notion of what is called associativity hold that x plus y plus z was x plus y plus z fine we also have the notion that 1 times x was 1 fine and there is this notion of distributivity that holds that this is same as alpha times x plus beta times x so we would like to take them as our definition of what are the set of vectors is that okay so against the last part will be something like x plus y times alpha is equal to alpha x plus alpha y so whenever writing alpha here alpha is an element of r or in general it will be an element of f fine because sometimes we may have complex numbers for us fine so the important thing to understand here is that whatever was true in r two or r three all right there we had vectors and those vectors had certain properties we want those properties to be satisfied for any set of vectors so we want the above properties to hold for any collection of vectors fine so for us definition is that definition let v be a non empty set set and if a set of scalars so remember scalars could be real numbers complex numbers fine it could be rational numbers it could be z p if you remember zp we had computed z 5 it could be z p p a prime number fine and the generalization of those ideas but mostly we will be looking at real numbers and complex numbers fine but it could be anything all right so let f be a non empty set and f a set of scalars then v is said to be a vector space over f so f does play a role that is important over f v said to be vector space over f if the above conditions hold above conditions holds means i have an addition between two vectors so given this set v given this at v i have a definition of addition between two elements i have a definition of a scalar multiplication there and with that all these conditions are valid that's more important for us is that okay so we already know that r2 and r3 are vector spaces let us look at more examples examples and there are hundreds of examples but will not look at all of them with just a few of them all right for example one let us look at r itself the set of real numbers then i can add two elements of r so given any x belonging to r y belonging to r i can talk of x plus y i have x plus y i also have x plus 0 is equal to x x plus y is same as y plus x fine we also have x plus minus x as 0 and so on so all the above properties are satisfied here it turns out that i can take take f to be r itself because the real number can be multiplied with real number to get a real number i can also take can also take f to b q set of rationals fine because rational times a real number is a real number so the multiplication by alpha where alpha is a rational number is a meaningful object and hence i can say that the set of real numbers is a vector space over real numbers as well as it is a vector is space over rational numbers it turns out that these two are very very different objects fine over rational numbers will not be bothered about because out of syllabus for us but this is very very important idea when you go for higher mathematics or higher places all right so will not be bothered about you but this is very very important it leads us to the study of rationalization and so on and lot of complicated things two example two you have already seen r2 r3 for you we can also talk of rn n is any positive number for us so i can take it as this r n is a set of collection of elements of the type x one x two x n such that each x i belongs to r one less than equal to i less than equal to n fine r n so i can write this vector as x so generally i am not putting transpose here but mostly we will be looking at as i said vectors are always column vectors for us so it will look like this x1 x2 xn i will just write x in general similarly i will have y which will be y one y two y n i can define x plus y as x one plus y one x two plus y two x n plus y n and alpha times x as i can define it as alpha x one alpha x two alpha x n all right what is important is that this is the right way of doing things but addition and multiplication is in our hand it is in our hand we can choose these definitions in such a way that some that said that you are trying to study is a vector space all right it depends on us how to define these objects nobody is going to dictate it is you who has to decide which to take which not to take and why fine depending on different issues i'll take some more examples example 3 let us look at rn we have already looked at complex number c n so now here i will have vectors are going to be again x 1 x 2 x n each x i belongs to complex number one less than equal to i less than equal to n and again i can have the same thing x plus y is x one plus y one so on till x n plus y n and you have alpha x as alpha x one till alpha x n fine important alpha can be real number alpha can be complex number fine so what we are saying is that c n is a vector space over r c n is also a vector space over complex number c n is also a vector space over rational number and all of them they have different meanings they have got different ideas so you have to be careful about them what you want to study and be careful about it for example here in this case all right i times so i times if the vector 1 to 0 0 0 is not equal to i 2 i 0 0 0 does not make sense this important all right does not make sense why does not make sense basically because i is not a real number all right as vectors as vector this is a vector this is a vector these two are vectors they are fine but this scalar multiplication is not allowed a scalar multiplication with complex number is not allowed fine so this is important when you are talking of complex number c n as a vector space over r you are not allowed to multiply by a complex number you are allowed to multiply only with real numbers similarly here you are allowed to multiply only with rationals here you are allowed to multiply with everything allowed to multiply by complex numbers but there it is not allowed thats what you have to be careful about fine so whenever you are looking at things be careful thats all it says nothing more than that some more examples fourth we had looked at the solution system so suppose i have got a matrix a is m cross n we have looked at the system all x belonging to r n so that a x is 0 all right this was called the solution set of the homogeneous system homogeneous system fine so look at here so if what we have already seen that if x one is a solution so let me write this as v so if x one x one and x two or x and y let me write x and y so if x and y belong to v this implies that x plus y times a is a x plus a y fine x y belongs to v implies a x is 0 and a y is 0 and therefore this will together imply that this is 0 plus 0 which is 0 not only that a times alpha x is alpha times a x because alpha is a scalar say r i am looking at over r so r here so this is same as alpha times zero which is zero so therefore if you see we are able to define vector addition and multiplication and you can check that all the conditions for v being a vector space is satisfied so therefore v is a vector space vector space over r y over r because we looked at the system over r all right so this r is related with r here all right or this also related with assuming that this has real entries if you are looking at complex numbers it could have been over complex numbers fine i have this now let us look at this set of all three cross three matrices so i have a and b belonging to set off so remember m three of r means this means set of three cross three matrices with real entries fine so given two matrices a and b which are three cross 3 i can add 2 matrices i can talk of a plus b which will also belong to m 3 of r we have a 0 matrix for every a we have a minus a and we also have a scalar multiplication alpha times a also makes sense and this is a three cross three matrix itself so therefore i can think of this as a vector space over r similarly if in place of the entries being real numbers if it was complex numbers i can talk in terms of complex numbers also fine one of the examples that i have not given you is six just think of a vector with having just a single element v is to be zero all right so zero plus zero is zero alpha times zero is 0 0 is a vector here all right so this is also a vector space over f whatever f you want to take real number complex numbers rational whatever you want you can do all of that fine 7 let me define the set r of x r of x is set of all polynomials in x with real coefficients so any element here looks like a 0 plus a 1 x plus a n x to the power n and a i belongs to r fine 1 0 less than equal to i less than equal to n fine note there is no restriction on n no restriction on the power fine so this set will contain constants so this will contain so 1 belongs to rx pi belongs to rx e plus pi belongs to rx we also have 1 plus 3x belongs to rx x square plus 10 belongs to rx we also have x to the power 100 plus 1 plus x square belongs to rx so all of them they belong to rx so any polynomial that you can think of polynomial means finite sum there is no infinite sum here there is a finite sum all right so look at any polynomial polynomial means finite number of monomials finite number of terms fine so that is an element of r x so given one polynomial given another polynomial i can add two polynomials to get another polynomial so therefore rx makes sense the addition of vectors addition of polynomials makes sense i can also multiply by scalar so every number gets multiplied so for example 1 plus 3 x times 2 is nothing but 2 plus 6 x so you are just multiplying it component wise in some sense and therefore that also makes sense so set of all polynomials in x with real coefficients make sense and this forms a vector space over r r because the coefficients are real numbers all right so r x forms of vector space over r so if i replace the coefficient by complex numbers will again have complex numbers rationals and so on coming into play all right in the last two examples if you remember matrix multiplication is defined similarly here polynomial multiplications are defined fine so matrix multiplication is a meaningful object a is a 3 cross 3 b is a 3 cross 3 matrix so i could multiply a into b to get a 3 cross 3 matrix fine but i am not bothered about those things all right we are bothered about only addition and scalar multiplication it has some extra things let's not worry about those extra things fine we want to restrict ourselves to looking at only vector addition and a scalar multiplication that is very important for us fine another example fix n belong to natural numbers for example take n equal to 3 all right then i talk of r of i do not know what was my notation in my notes let me just look at it do i have a notation here x and n yeah this is notation so this is a zero plus a one x plus a two x square plus a three x cube a i belongs to r zero less than equal to i less than equal to three fine so here we are fixing degree of the polynomial is less than equal to three all right so if i have two polynomials of degree less than equal to three then i can add those two polynomials and the degree of the polynomial is not going to increase because i am not multiplying this is important since i am not multiplying i am just adding two vectors so i have got i supposed to add only two polynomials of degree three or degree less than equal to 3 therefore it makes sense fine similarly scalar multiplication makes sense so this is again a vector space vector space over r fine note if we collect vectors collect polynomials polynomials of only degree 3 then we have we have a problem what type of problem do i have all right so let us take a polynomial x three plus one take another polynomial which is minus x cube plus x plus one these two are polynomials of degree three if we add eight fine this x cube x cube cancels out i am left out with x plus 2 not a polynomial of not a polynomial of degree 3 all right so there is a problem is that ok so therefore when you want to talk of polynomials we have talk in terms of polynomials of degree less than equal to some number so that is n that we are talking about here so you can fix it and then proceed fine now one very important example you may be saying that what is new about it we are looking at the same things again and again that's true most of the times we'll be looking at the same polynomials or same matrices same vector same rn and things like that but there are places where things are different so let me take an example here so that example was eight let me look at example nine i am going to look at this at r plus r plus is set of all alpha belonging to r so that alpha is positive all right this is my set this is a non empty set set take any two elements here alpha and beta i want to define the sum as not alpha plus beta but something else so i put a plus sign here to say plus is this is not the usual real number sum real numbers are sum differently this is not the same all right so here i am saying that this is i define this to be equal to alpha into beta so what i am saying is that 1 plus 2 is equal to 1 into 2 which is 2 1 upon 5 plus 1 upon 5 is one upon five into one upon five which is one over twenty five is that ok all right the addition the vector addition that i am defining it has changed fine so here for example in this example this example 0 does not belong to r plus so i cannot talk of 0 plus something fine but i can talk of 1 plus in in real number alpha and this by definition is equal to 1 into alpha is alpha so this implies 1 or this is same as equal to alpha into one which is same as alpha plus one so one is the additive identity so one is the one which works as something nice fine also alpha plus 1 upon alpha is alpha into 1 upon alpha which is 1 so the inverse the additive inverse is 1 upon alpha of alpha fine it makes sense because alpha is a positive number here alpha is a positive real number fine i have defined a scalar multiplication so let me take a as a scalar fine and alpha as vector fine i can talk of this as defined to be equal to alpha to the power a alpha is a positive real number a is a scalar real or com real number so whether it is negative or positive alpha to the power a makes sense all right so a can be our real number a could be 0 also so any number to the power 0 is 1 if a is positive number you'll have positive powers if a is negative so a negative will imply i am looking at 1 upon alpha to the power a alpha to the power a will correspond to so a is negative so this is equal to say minus t so this will be alpha to the power t fine t positive so accordingly you can make sense and everything is done so you can show that r plus is a vector space over r again you can make it a vector space over rationals and over anything but important this is a different vector space then all the vectors that i have looked at fine there are other vector space also tenth i will just give you the last one i can think of this space q of root two and root three so this is the collection of all elements of the type a zero plus a one root two plus a two root three plus a three root six and a i belonging to rational numbers so you can see that you can add two vectors you can multiply by a rational number so this is a vector space over q fine so will be looking at all the examples till eight will also look at some more examples future whenever we come across them these two ninth and tenth are given to you to make you understand that just looking at plane r2 r3 rn and items coming from there is not enough fine these are also very important ideas which give us different problems to solve at to look at and that they have got different applications for example this will tell us that how do i rationalize numbers so can i talk of we used to talk of 1 plus root 2 multiplied by root 2 minus 1 by root 2 minus 1 what do i mean to say that i can look at 1 plus root 2 plus root 3 plus root 5 and so on here we don't have root 5 i can just include root 5 root 10 and so on and you can look at things fine so that's all for now we'll look at vector subspaces linear span and things like that in the next class thank you so [Music] you