this is chapter 4 section 1 vector spaces and sub spaces and in this video we're just going to look at vector spaces but I just want to caution you that's starting with this chapter things are going to shift a bit to the more abstract which means it's going to be a little harder to wrap your mind around things you might have to put in some extra time into studying into trying to understand the examples I'll do my best to give you all of the material in the videos but you may find yourself referring to the textbook more often or finding other videos to supplement and if you do certainly let me know I can embellish these videos a bit more but I try to keep these short enough to give you the main idea without you know making a 45 minute video so consider yourself warned it's going to be a bit more difficult moving forward but I know you can do it so what is a vector space it is a non-empty set of vectors and two operators which is which are addition and multiplication such that the following axioms hold for all vectors u V and W that belong to the set of vectors and again we're just saying they belong to the set and for C D which are scalars that belong to the set of real numbers so before we talk about each of these axioms I want to point out to you that we've actually kind of already done this so as we go through the axioms you'll say okay I know that one yes we've done this one yes we've done this one but what we've done up to this point is simply use them for the set of RN so RN is a vector space and all of these axioms hold true which is why all of these will look familiar to you is because we've been here we've been typically in r2 or r3 some times in our four or five etc but what we want to do now is look at different spaces and we need to verify these axioms to see if those other spaces may in fact be vector spaces or not so let's take a look at the axioms and try to make some sense of them the first one simply says that if you have two vectors then the sum of those vectors should also be in the set V so of course this holds true for r2 and r3 and again we're going to look at some examples where that might not hold true it also shows that the associative and commutative property so commutative here associative property here that doesn't look like an a associative property here hold true for addition it also says that the zero vector so I'm going to put a box around this one and you'll see why in a little bit it also says that the zero vector needs to be in our vector set our vector space where zero the zero vector plus any vector is equal to that vector so this is essentially the identity and this is the inverse the inverse saying I can add something and again each inverse is unique that I can add some inverse to an existing vector both are in the vector space and the sum of those two vectors is zero so those are all of the addition ones now let's take a look at multiplication we again now have that a scalar times a vector is in the vector space so I can take the scalar times something whoops I'm gonna put a box around that one and then we have the distributive property that I can distribute a scalar that and this is similar to the distributive property again just on the right side instead of the left side but it's distributive property this one is the associative property and again it's associative with the scalars so it's not associative we know that matrix or vector multiplication is not associative or commutative but of course if we're dealing with scalars those are and then back here again we're down to the identity so as you can see I've put a box around a few of these and I put the box around them because those are the ones that we will most often find issues with so those are the ones that you're probably going to want to check first so let's take a look at a few examples of where we might not have a vector space let's take a look at an example which may or may not be a vector space and I have purposely started with some examples that we could visualize as I said we're going to get more and more abstract so the examples that we'll be able to visualize will be fewer and further between but here's a great example we can visualize I have set V of all of the vectors X Y such that X is greater than or equal to 0 and Y is less than or equal to 0 so I know because I've been graphing for quite some time that if X is greater than or equal to 0 and Y is less than or equal to 0 that's this space down here that's the fourth quadrant so my question is if I have a vector in the fourth quadrant will it meet all ten axioms and in fact be a vector space so let's take a look at an example let's say vector U is 3 oops I almost wrote that as an ordered pair 3 negative 1 so 1 2 3 negative 1 here's vector u thinking about those ten axioms and there were three that I said we're kind of the most important to check or the ones that you would check most often where things wouldn't necessarily work out in your favor let's start by looking at axiom 1 which says the sum of U and V is in V so I only have you but let's take V and obviously U is in V already but let's give myself a vector V to just C so vector V let's say is 1 negative 4 and again this is also in V and I can see that because X is greater than or equal to 0 and Y is less than or equal to 0 so I wouldn't have to have the picture to show that they are in V but my question is is the sum of those also in V so let's just see well we know the parallelogram rule says so visually I can just look at this and say yes that was a really bad parallelogram but you get the idea but I can also add u plus V which would give me 3 plus 1 or 4 and negative 1 plus negative 4 which is negative 5 and 4 negative 5 is in fact in V because X is greater than or equal to 0 and Y is less than or equal to 0 so this one's okay what's the next one I said to check number 6 just kidding number 4 number 4 that says that the 0 vector is in V so looking here this is or equal to 0 or equal to 0 so that means the 0 vector is in fact in V and the last one which is six which says see you or CV I don't remember which letter I use hey quit changing colors on me CV is also in V and keep in mind that C is just some real number any real number that I want and this one hopefully you can see is going to be a problem because what if C was negative three then C V would be negative 3 times 3 negative 1 which gives me negative 9 3 which is somewhere over here which is clearly not in my region of X is greater than or equal to 0 and Y is less than or equal to 0 so negative 3 V does not belong to V and therefore this is not a vector space so notice I didn't say it doesn't meet axiom 6 because remember your professor doesn't have all the axiom numbers memorized I have the concepts down that's what you should do too is don't rely on those numbers but rely on logic so my logic says that I can't take all C V and have it belong to V this one is not met and therefore it's not a vector space here's another practice for us that looks very similar and if you'll notice the only thing I changed as I said that both are greater than 0 so you might be thinking well duh brain we get it that this in fact will still be the case that I cannot take some vector that's in the space that I want where both are greater than 0 I can't take it times anything that I want and still end up in that same region and that is true but I why don't you give you this example specifically because this one also is that zero is not the zero vector which is 0 0 is not envy because this is greater than zero and greater than zero so zero is not included so again obviously we already knew based on this that it was not going to be a vector space but the fact that the zero vector is not included also tells us that V is not a vector space here's a practice for you to try on your own so give this one a try when you're ready press play to see how you did so again this is one that we can visualize and we are saying that X Y is less than or equal to zero which means X Y is negative or equal to zero and that would mean that either X is positive Y is negative or X is negative Y is positive so here's what we're dealing with so let's check a couple of those axioms that have a tendency to give us a hard time for instance is the zero vector in V well because it's or equal to zero yes in fact that is true I could have V as zero zero which is the zero vector what about C V is that in V as well so quite often this is one that gives us a hard time for instance let's say this is 2 negative 3 we'll say V is 2 negative 3 this certainly is in V because 2 times negative 3 is negative 6 and that is less than or equal to 0 and what if I took that times a positive value well that would give me 6 negative 9 which is still in that same region ok it's just going to make this guy longer what if I took it times negative 3 well that would give me negative 6 positive 9 would that still work well if I multiply - that is still less than or equal to zero and in fact it would just be over here and so yes we are just fine this still holds true so check check I've got two checks now let's go back to that very very first one that says u plus V must be in V well let's say I'm gonna get rid of these green guys here let's say that's my first one and we're going to make I'm gonna get rid of this running out of room here so let's say that U is negative 1/5 so negative 1/5 up here and again what is then u plus V I would end up with 2 plus negative 1 which is 1 and negative 3 plus 5 which is 2 and 1/2 which again parallelogram rule we could see clearly this guy is not in V and therefore no because this guy is not true therefore this is not a vector space let's take a look at polynomials polynomials in fact is a vector space which seems a little counterintuitive but we can see in the way that this is written that it's written very much like a linear combination of scalars and vectors same idea and remember that the degree of a polynomial is the greatest power with a nonzero coefficient so in this example n would be the degree of my polynomial assuming that a n is not 0 and the degree of a constant is 0 so for instance if my polynomial oops I went ahead and wrote it down here so if my polynomial was just equals a sub zero meaning I don't have a tea or a tea squared or a tea to the n then that is the zero degree and if all terms have degrees zero then obviously the highest degree would be zero and that's called the zero polynomial so let's verify the axioms to ensure that this is in fact a vector space so I've listed the axioms here so that you don't have to have them memorized but let's take a look at the first one that you plus V would in fact be in the vector space so we know the vector space essentially are terms that look like this they're going to be a polynomial written in this form so if I added say P and Q together so I've written these in terms of vectors but if I added P and Q together would it in fact be in that space so if I take P of T plus Q of T what would I get I would get a0 plus b0 and then I would get a1 plus b1 T because I'm adding like terms plus etc etc and that pattern would continue and would that in fact be a polynomial yes I would just be combining those like terms so this one does in fact hold now what I know about real numbers is that addition of real numbers is commutative and also associative so I'm not going to check those because I know that if these guys were in the opposite order that would still be okay then if I were grouping things differently I'm still combining like terms and that's okay so those have been checked and verified is the zero vector well that means oops P of T what if P of T were zero would that fit this yes that would just be the a of zero was zero is that in the space yes it is and if I added this to any other polynomial what I end up with a polynomial yes I would what else can I check let's check the inverse so P plus negative P well it would certainly make sense if I had a zero and let's say Q of T were negative a 0 and then this was a 1 T then this was negative a 1 T and we can see if I added those together I would in fact get 0 so cheked cheked let's also check if I multiply by something well same as our example here and again I keep erasing I apologize so there's only so much room let's say I take 3 PT well then it's just 3 a 0 plus 3 a 1 T 1 etc would that fit a polynomial you betcha and again these all follow once I know this one is true these are just properties of real numbers that can be checked very easily so hopefully I've given you enough to go on there that polynomials are in fact a vector space