In this video, we will learn what real vector spaces are, and we will see some examples of real vector spaces. So, what are real vector spaces? See, we have two things here, first of all I set V, right? This V we were calling it vectors. These vectors are actually maybe set of matrices, maybe set of polynomials or maybe a set of real numbers, right? And next is set of scalars, right? So, here I am taking set of scalars as all real numbers, a set of scalars, right? So, when this set V is said to be a vector space, right, or real scalars, right? So, a non-empty set of vectors or a set of real numbers with two operations, a star and dot, right? So, here we define two operations star and dot. With these operations, it is said to be a vector space if it satisfies following properties. Now, what are these properties? Respect to star, it’s first of all, it should be a binary operation. So what do you mean by binary operation? Binary operation means that you take any two elements of the set V, apply operation star, the resultant must be in V, right? The second property is respect to star, it must satisfy commutative and associative property, right? The third property is, that there should exist one element e in V such that e star V is equal to u star e, equal to u for every element u in V. We will understand these concepts by an example. First of all, we will see that what vector space are. And then we will illustrate all these properties by an example, right? Now, for each element u in V, there should exist an inverse element V in u, V in V, such that u star v is equal to e is equal to v star u, right? And then first of all respect to dot, dot is an operation from R cross V to V. That means the first element is from R, the second element is from V, right? When you apply this dot operation, the resultant element must be in V, right? And respect to dot, respect to star and dot, it satisfies these four properties, right? The first property, second property, third, and fourth. We will see these properties by an example, that will be clear to you. So, the example, let R+ be the set of all positive real numbers. I mean what is R+? I have defined R+, I have taken all x in R, right? All real numbers, such that x is greater than zero, right? Then define the operations, addition, scalar multiplication as follows. This star, I'm calling it addition, right? So, I have defined this operation as u into v, use your multiplication. And this dot as u raised to power alpha, right? For all u in R+ and for all real numbers alpha. So, this will constitute a vector space under these operations. How this will constitute a vector space? So, let us see one by one. The first property is, that star must be a binary operation. How is a star defined? A star is defined by the usual multiplication, right? Now, when you take two real numbers, when you take two u, v belongs to R+, right? I mean, R+ means positive real numbers. If you multiply them, right this is operation. How operation is defined by usual multiplication. And the product of two real numbers again, two positive real numbers is again a positive real number. So, it will belong to V, right? V here is R+, right? So that means it is a binary operation. The second property is, it must be commutative. So, we can easily see that if you take u star v, which is equal to u into v, right? That is same as v into u because it's a usual multiplication, and which is equal to v star u, right? So that means it satisfies commutative property. And similarly, we can see that, if you take u star v star w, right? Which is equal to u, star vw by the definition. And this is equal to u into vw, right? Because this star is nothing but multiplication of these two elements, right? Which is equal to u into v with w because usual multiplication satisfies associative property. So, it is equal to u star v with w, right? And this is equal to u star v star with w. So that, in this way we can say that this is equal to this, that means they satisfy associative property also, right? Now the identity element, now, if you see, if you take e star u, right? This is equal to eu, right, by this definition. And that must be equal to u, which implies e equal to one for every u. So, you can easily verify that if you multiply any element with one, it will be itself always for every v, right? So, the identity element here is one, which exists and which belongs to R+, right? Then the next property is inverse element, right? Now, if you take u belongs to this R+, and if you take v element, right? Which I am talking about the inverse of this u, then u star v is equal to e, e is one. This by definition is uv is one, then v is nothing but 1/u. So, if you take inverse of any element, say inverse of two, right? Inverse of two is one by two, inverse of three is 1 by three, and which belongs to R+ which also exists, right? So hence we have seen that all the property respect to star holds. Now this respect to multiplication or dot, right? Now if you see the dot, how dot is defined? Dot is defined in this way, we have to see that what the definition by which star or dot is defined. And we have to apply, we have to see whether the properties which we have stated by holding for these operations or not, right? First of all, the first property here is, this property that R cross V should go to V. That means if you take any element of this R and any element from V should give a result that element should be V itself, right? So, here, how this is defined. You can see that r dot u is u raised to power alpha, right? Now, if you take any real number, may be negative or positive, right? Say you take minus one, right, minus one dot of u. Which is u raised to power -1, right? And from where u is coming, u is R+, right? So, this is always in R+, there's always a positive real number, right? So, I want to say that if you take any real number negative or positive and any u in R+, the resultant element which is defined by u raised to power alpha is also is always in R+, right? So, we can say that, the first property respect to dot is satisfying. Now, let us try to see some other properties. See alpha dot beta dot u, must be equals to, you can see here, alpha dot beta dot u should be equals to alpha beta dot with u, right? Now, this we have to verify, for example, so you take the left-hand side. What is left-hand side, alpha dot, what this bracket means, u raised to power beta by the definition. And this means what? This means u raised to power beta raised to power alpha, right? So, this is equal to what, u raised to power beta alpha, right? Or u raised to power alpha beta. So, if you take this as a real number and apply this definition over here, then this will be what? Alpha beta dot with u, which is same as, right-hand side. So, hence this property holds, right? Now, if you take one dot u, one dot vu is what? U raised to power one, which is u, so this property is also satisfied. Now, if you take alpha dot u plus v, plus means a star, right? Is uv is equal to, so it should be equal to u dot alpha dot u star with alpha dot v, right? Now, this is, this side, if you see this is alpha dot, this is multiplication by the definition and this is equal to uv raised to power alpha, right? Now, see the right-hand side, this is what? U raised to power alpha star with v raised to power alpha. And if you apply the star property, then this is nothing but u alpha into v alpha, which is equals to uv hold this to power alpha. So, we can say that both are equal, right? So, we can say that the left-hand side is equal to right-hand side, this property holds. So similarly, one can also show that this property holds, on the same lines we can easily show, right? So hence we can say that, if we define the operation is star and dot in this way, for the set of positive real numbers, then it will constitute a real vector space, right? So, in this video, we have learned that what real vector spaces are. We have also seen one example to demonstrate that, real vector spaces, how we can prove that a given vector space under the given operations constitute a vector space. [MUSIC]
