 So, in today's lecture we look at the notion of a vector space almost all of us have done vector calculus, calculus involving two variables, three variables. For example, look at vectors in the plane, we have done vector addition and then multiplying a vector by a scalar. For example, this is called a linear combination of vectors. We have also encountered linear combination of functions especially when we have discussed the notion of solutions of differential equations. If y1 and y2 are two solutions of a linear differential equation then a linear combination alpha times y1 plus beta times y2 is also a solution of the differential equation. The structure that formalizes this notion of linear combination etc is called a vector space. This is a basic object of linear algebra. So, what I will do is today give the definition of a vector space and then give a variety of examples just to tell you that vector space is abundant in the whole of mathematics. So, these examples for instance will include the differential calculus example that I mentioned and also other examples. So, let us look at first the notion of a vector space. So, here is the definition. A vector space consists of the following, consists of the first object as field. So, we say vector space over a field. The first object is f, f field. We say that it is a field of scalars. In this entire course of linear algebra we will restrict our attention to mostly r and in some examples the complex field also. Then we have v, a set of elements, v of course is non-empty. Set of elements, these elements are called vectors. These elements are called vectors. Then third aspect is a rule called vector addition. This is a binary operation which for x, y in v assigns a unique vector which we will denote as x plus y. This x plus y belongs to v. So, there is a binary operation on v. It is called the vector addition. This binary operation satisfies the following conditions. For every x, y in v there is a unique x plus y which also belongs in v. So, it is close to the respect to this operation such that the following conditions are satisfied. Such that let us write these conditions here. First it is commutative. x plus y equals y plus x. Second condition it is associative. x plus y plus z equals x plus y plus z. So, that is associativity. Condition 3, there exists an element denoted by 0. That is in v such that, so this 0 acts like additive identity. So, we have this condition to be satisfied such that x plus 0 equals x. See, all these are for all x, y, z in v. So, such that this holds for all x in v and the last condition, fourth condition is that for every x in v there exists an element which we denote as minus x. This minus x belongs to v such that this is like the negative additive inverse negative element such that x plus minus x equals the additive identity. So these conditions hold for all x, y, z in v. So, this is with respect to the vector addition operation these four conditions must be satisfied. So, we have vector space has a field. So, there is a field over which the vector space is defined. There is v a set of vectors satisfying these four conditions. Now what is the interaction between the field elements and the vector elements that is given by scalar multiplication. So, I will say for a rule called scalar multiplication, a rule called scalar multiplication which for every x in v and a scalar alpha in f which for every x in v and alpha in f assigns a number alpha times x in v. So, for every x in v, alpha in f this product I mean the scalar multiplication alpha times x must be in v and this scalar multiplication must satisfy the following four conditions. So, what are those conditions? Remember that the underlying remember that f is a field. So, f for instance has 0 as well as 1, 1 is the multiplicative identity. So, the first condition relates use as the multiplicative identity. So, I will again write four conditions for the scalar multiplications 1 into x equals x. So, for all x y in v and for all alpha beta in f. So, I have this condition 1 into x equal to x condition 2 alpha into x plus y equals alpha x plus alpha y. I should actually write a dot here but we will follow the convention of not using this dot. Alpha into x plus y is alpha x plus alpha y condition 3 is if you look at alpha beta of x it says you do it repeatedly. This is alpha into beta x and finally condition 4 is you take the scalar alpha and beta look at alpha plus beta of x thus this must be alpha x plus beta x. So the scalar multiplication must satisfy these four conditions the vector addition must satisfy these four conditions the underlying set f must be a field then v is called a vector space over f. So, in this case we say that v plus dot is a vector space over f. So, this is a formal definition of a vector space alright. Let us look at some examples now before we proceed with the examples let us understand that the two examples that I told you in some sense as motivating examples satisfy these conditions example of two solutions coming from a differential equation and the behavior of vectors in the plane for instance. So, let us look at some examples I will give an abundance of examples just to illustrate that this is an absolutely fundamental object in mathematics not just in linear algebra. So, let us look at examples let us start with the simplest example you take v as R also the underlying field as R then we know that then addition is what x plus y is the usual addition usual addition of real numbers scalar multiplication is also the usual multiplication then this is a vector space v over f is a vector space that is we say that R is a vector space we will follow this notation only for this today's lecture we will simply say R is a real vector space C is a complex vector space etc. So, R over R is a vector space as I said we can also say that R is a real vector space. So, this is the basic example similarly one could define C as a vector space over itself in general any field over itself is a vector space over that field second example is also trivial take v to be single term 0 and f is any field you have to define C must define vector addition scalar multiplication vector addition is defined like this scalar multiplication alpha times 0 equal to 0 for all alpha and f then this is a vector space v over f is a vector space over the field f okay these are trivial examples let us look at the first non-trivial example look at f 2 f 2 is what f 2 is the set of all set of all vectors x written in this manner x equal to x 1 x 2. So, I will write it as a column vector set of all x 1 x 2 such that x 1 and x 2 both belong to the underlying field. So, this is f 2 okay set of all column set of all columns which set of all columns which have just 2 coordinates which have just 2 components what is vector addition scalar multiplication once I specify that we can verify whether it is a vector space. So, let us take 2 elements x y in f 2 then x can be written as x 1 x 2 y can be similarly written y 1 y 2 let us also take alpha in the underlying field then x plus y we define x plus y to be the column vector the natural way to define addition is x 1 plus y 1 that is the first coordinate x 2 plus y 2 is a second coordinate this is obviously a binary relation scalar multiplication alpha x you do it for each coordinate multiply the scalar alpha to each coordinate alpha x 1 alpha x 2 then it is an easy exercise to verify that f 2 over f that is v equals f 2. So, v over f is a vector space with respect to this addition operation and this scalar multiplication okay now we can do a little more general we can do a little more general. So, let us look at f n f n this time is the set of all x that can be written as x 1 x 2 etc x n. So, there are n coordinates this time such that each x i comes from the underlying field this simply extends f 2 that was given in the previous example how do you define addition as before coordinate wise x 1 plus y 1 x 2 plus y 2 etc x n plus y n. So, x is this y is y 1 y 2 etc y n addition is defined in this manner scalar multiplication similarly for alpha in the underlying field then again we can show that f n over f is a vector space with respect to these operations okay. So, this is a little extension of the third example we can do a little more general let us look at f m cross n f m cross n now this m cross n should suggest the objects that we are considering here. So, what is the definition here this is the set of all x such that x has m into n coordinates that is one way of looking at it which will give rise to the same vector space as above a slightly different way of looking at f m cross n which is the set of all a which can be written as a 1 1 a 1 2 etc a 1 n. So, this is a set of all m cross n matrices matrices with entries coming from the underlying field which matrices which have m m rows and n columns a 1 etc 1 1 etc a 1 n a 2 1 a 2 2 etc a 2 n let me write the last row a m 1 a m 2 etc a m n. So, these numbers a i j they all come from f 1 less than or equal to i less than or equal to m 1 less than or equal to j less than or equal to n such an object is called a matrix. So, you collect the set of all matrices with the property that the matrices have m rows and n columns the entries of the matrix come from the underlying field f then again the operations of vector addition and scalar multiplication is natural by the way here it is addition of matrices but still we refer to it as vector addition that is each element in f m cross n will be referred to as a vector okay now this is the terminology that we will adopt. So, a vector will not denote any more objects on the plane or objects on the three dimensional space they can now denote matrices we will do a little more general we will use vectors to denote polynomials we will use vectors to denote solutions of differential equations etc okay. So, let us look at vector addition scalar multiplication. So, take two matrices a and b a return in this manner b return in a similar manner then a plus b is coordinate wise addition. So, a plus b is a 1 1 plus b 1 1 a 1 2 plus b 1 2 etc a 1 n plus b 1 n a 2 1 plus b 2 1 etc a 2 n plus b 2 n similarly the last column last row a m 1 plus b m 1 etc a m n plus b m n. So, this is how vector addition is defined scalar multiplication as before alpha times a will be you multiply each entry by the scalar alpha alpha a 1 1 alpha a 1 2 etc alpha a 1 n alpha a 2 1 alpha a 2 2 etc alpha a 2 n etc a m 1 a m n. So, this is how scalar multiplication is defined again one can verify that f m cross n over f is a vector space. See for instance the 0 element will be the 0 matrix the matrix all of whose entries are 0 the negative element of an element a will be the matrix whose entries are the negatives of the corresponding entries of the matrix a etc. So, this is another vector space example of vector space let us next look at p n r for instance what is p n r p n r equals the set of all polynomials the set of all polynomials in a real variable in a real variable t with degree of the polynomial not exceeding with the degree not exceeding the number n that is used as a subscript as the subscript here. So, it is set of all polynomials of degree less than r equal to n polynomial in what polynomial in one single real variable t that is what is denoted by this r ok. Now for example if p q belong to p n of r then these are polynomials I can write polynomials in the following manner p can be written as p naught plus p 1 p plus p 2 t square etc plus p n t to the n remember some of these constants p naught p 1 p 2 etc p n some of these constants could be 0 because it is a set of all polynomials of degree not exceeding n the degree could be less than n in which case p n is 0 for instance similarly q q naught plus q 1 t plus q 2 t square etc t is real variable. So, I have two elements two vectors now so vectors we are using the name vector for a polynomial here. So, what is vector addition p plus q that is again the natural addition p plus q is defined as so definition is p naught plus q naught plus p 1 plus q 1 t plus etc plus p n plus q n t to the n. So, this is a definition of addition vector addition scalar multiplication also you would simply bring it to the coefficients. So, alpha times p is alpha p naught plus alpha p 1 t etc plus alpha p n t power n then you can again verify that v is this p n r is a real vector space okay. So, p n r is a real vector space vector space over the field of real numbers. Let us do a little more general let us now look at the space of continuous functions the space of real valued continuous functions on a closed interval a b the space of real valued continuous functions on the closed interval a b we should again so I am saying the set of all real valued to continuous functions. So, the underlying field will be r we should again define vector addition scalar multiplication now that is similar to what we have done earlier for the case of polynomials this is so called point wise addition. So, we will do a similar thing here for p q in c of a b define p plus q of t. So, p plus q is a new function whose definition is p plus q at the point t is p of t plus q of t and scalar multiplication is defined similarly alpha p is a new function it is just alpha times p of t is most natural way of defining it for all the alpha in r for p q in c of a b. So, it can again be shown that c of a b is a real vector space if you replace the underlying field by c and then consider complex valued continuous functions then c of a b will be a complex vector space we can do a little more general. Let us denote c k this time open interval to be the set of all functions real valued functions. Let me say f with the property that let us say that I will use the variable t for any point on the interval a b. So, it is f of t for instance with the property that d k f by d t k the kth derivative of f is continuous the property that d k f by d t k the kth derivative of f with respect to of course there is only one variable here the independent variable is t this is continuous in a b collect all such functions real valued functions having this property again vector addition scalar multiplication as before vector addition and scalar multiplication as before then this is a vector space is a real vector space we can do a little more general one looks at c infinity of a b this is now the set of all functions real valued functions that are infinitely many times differentiated in the interval a b open interval a b same operations of vector addition scalar multiplication will tell you that this is a real vector space that c infinity a b ok that c infinity a b let us consider a more let us consider another example I will give it as example 10 let us consider let me use this notation ok let me use the notation f of let me use the notation f of a b a b, r to be the set of all functions f from a b to r set of all real valued functions depend on the open interval a b collect all those functions that is my capital f a b, r now in this look at v as the set of all f in f such that integral a to b f of t dt exists that is f is Riemann integrable that is set of all functions that are Riemann integrable then with respect to again usual addition and scalar multiplication we can show that v is a vector space then v is a real vector space now what is the meaning that this is a vector space if f is Riemann integrable g is Riemann integrable then f plus g is Riemann integrable if f is Riemann integrable alpha is a real number then alpha times f is Riemann integrable ok example 11 suppose that we have a matrix a of order m by n so there are m rows n columns entries are real numbers define v as the set of all vectors x and r n such that a x is 0 a x equal to the 0 vector now remember a is an m cross n matrix x is n cross 1 any vector standing alone is a column vector so when I write a vector as it is it is a column vector so the product here is m by n into n by 1 so the usual product of matrices so the resultant is m cross 1 this so this 0 vector on the right hand side is a 0 vector which has m coordinates now this is a vector space with now this is a subset of r n the subset of r n r n already has vector addition scalar multiplication defined it can be shown with respect to those operations that v is a vector space again close with respect to addition means what if x and y satisfy ax equal to 0 and ay equal to 0 then x plus y satisfies a of x plus y equal to 0 because a of x plus y is ax plus ay similarly if x belongs to v then alpha times x also belongs to v because alpha comes out here of this equation alpha comes out of this equation so this is a vector space we also have another example this the example that I gave as a motivating example let us look at the operator L defined on a function y which is at least n times differentiable y is a function of the independent variable let us say t define L of y so I am looking at this equation L of y equals L of y is defined by d to the n y by d t to the n plus a 1 d to the n minus 1 y by dt n minus 1 plus a 2 d n minus 2 y dt n minus 2 plus etcetera plus a n minus 1 plus a n minus 1 d y by d t plus a n y of t let us so let L be defined in this manner L is a differential operator consider v as the set of all y such that set of all y such that L of y equals 0 okay see I must mention here that a 1 etcetera each of these is a function of the independent variable t each of these is a you can take them as constants in particular but in general they are functions of t then look at v as a set of all y so this you need y to be at least n minus you need y to be at least n times differentiable in order for this to be defined so collect all those functions then that satisfies then collect all those functions then such a set is a vector space if you consider real valid function then it is a real vector space let me give you a couple of more examples some intuition coming from geometry so example 3 v is the set of all x in R2 so this is a subset of R2 in fact it is a vector space in its own right set of all x in R2 such that x2 equals alpha times x1 alpha is a fixed number x2 is alpha times x1 alpha is a fixed number okay so which means you collect the set of all vectors which have the property that the second coordinate is alpha times the first coordinate okay geometrically what does this set represent this set represents the set of all points passing through the origin this set represents the set of all points lying on a straight line passing through the origin okay it can be shown again that this is a vector space the set of all points lying on a particular straight line passing through the origin the slope is given by alpha a little more general I will call that as example 14 v is the set of all vectors in the 3 space that satisfies something like x1 plus 2 x2 plus 5 x3 equals 0 geometric interpretation this is the set of all points lying on a certain plane passing through the origin okay this can again be shown to be a vector space okay so I hope the abundance of these examples illustrate the importance of the notion of a vector space let me conclude the lecture by giving 2 by giving at least one example of a subset of R2 which is not a vector space example 15 let us look at v as the set of all x in R2 such that x2 is x1 plus 1 x2 is x1 plus 1 the second coordinate is you add 1 to the first coordinate collect all such vectors this is again a straight line the only thing is that the straight line does not pass through the origin you can show that this is not a vector space okay now you must observe that vector space must have 0 0 must belong to the vector space now this one does not have 0 okay so this is not a vector space one final example non example so this is not a vector space one final example let us say v equal to the set of all polynomials set of all polynomials of degree precisely 2 I do not say degree less than or equal to 2 degree precisely 2 I leave it for you to show that this is not a vector space it is not even close with respect to addition v is not a vector space okay okay let me stop here in the next lecture I will discuss the notion of subspaces examples of subspaces etc.