 One of the central ideas of linear algebra and of higher mathematics is the concept of a vector space. So what's a vector space? Well, suppose I have a set of vectors v and a scalar field f. We say that the vectors form a vector space over f or a f-vector space provided that all of the following conditions are met for all vectors u v w in our set of vectors and all scalars a b c in our scalar field First we have to have closure under vector addition when I take two vectors v and u Their sum is also going to be one of the vectors in our set Next we have to have closure under scalar multiplication If I take some scalar c from our field f Then the scalar multiple c times my vector has to also be an element of my set of vectors Third we have to have commutativity of vector addition v plus u has to be the same thing as u plus v Next vector addition must be associative if I add three vectors It shouldn't make a difference whether I add the last two first or if I add the first two first I should get the same result But wait, there's more We also want to make sure that we have a zero vector So without committing to what it actually looks like there should be some vector zero that for any other vector that we choose The vector v plus our zero vector should just give us back the vector v Tied very closely to that is the existence of an additive inverse for any vector v in our set There should be a vector that will designate as negative v Also in our set for which the sum of the two vectors is zero Next while we have defined scalar multiplication by an element of f We haven't committed ourselves to what happens when you multiply by the scalar one Which is the multiplicative identity in our field and if we are dealing with a vector space We do want to require that the scalar multiple one times any vector should give you the vector itself What about scalar multiplication? Well, I can't scalar multiply two vectors That doesn't make any sense but what I can do is I can multiply a vector by a scalar and another scalar and so I would like to make a Requirement for a vector space that we have what you can think about as associativity of scalar multiplication that a times the vector b v is the same as a b times the vector v and Then I also want a distributive property of scalars over vectors now This actually comes into flavors one possibility is I might have a scalar multiplied by the sum of two vectors and I would like if we're dealing with a vector space for that to be The scalar times the first vector plus the scalar times the second vector now Notice that what we've done is we've taken the thing on the left which is our scalar and Distributed it among the things in the parentheses and because we've taken the thing on the left and Distributed it this property is sometimes called left distributivity And at this point you might ask yourself self could I have right distributivity and The answer to that is a fairly carefully constructed. Yes, and in this particular case what it means is that we're going to Distribute a vector over the sum of two scalars And so we have our second distributive property which corresponds what you might think about as a right distributivity if I have a sum of scalars Multiplied by a vector what I'm going to get is the sum of the scalar multiples of the vector and So what we have are 10? Count them 10 properties that are required if we're going to be a vector space And you may want to think about it this way a vector space is sort of a very Exclusive club that only certain types of things can join to join club vector space You have to meet all 10 requirements So let's see if we can find some of these members of club vector space Well, let's start out with a familiar set How about the set of integers and we do have to define what we mean by addition and scalar multiplication? So let's go ahead and just define them in our usual terms where addition is our usual addition and Scalar multiplication is our usual multiplication So let's see if the set of integers has what it takes to become part of a real vector space So remember that the real part of the definition refers to the field over which we're going to be conducting our operations All right, so the membership committee for club real vector space receives Z's application and They go through the checklist So first of all we have to consider any three elements of our set So let's take three integers mn and p and we'll check out each item So is the set of integers closed under addition if I take two integers Do I in fact get another integer and the answer to that is yes All right now. How about that closure under scalar multiplication? So for a real vector space these scalars are going to be drawn from the set of real numbers and The membership committee requires that the scalar multiple also be an element of the set but if C is a real number but not an integer then The scalar multiple C times an integer is not going to be an integer Which means that the integers fail the second requirement Sorry, Charlie. You do not have what it takes to form a real vector space