 In this final video for section 1.3 about vector spaces, I just wanted to mention some elementary properties for vector spaces. So for any vector space V whose coefficients, whose scalars come from the field F, we have the following five properties. And these are very elementary properties, many things we take for granted here. And so I want to mention that if we have a generic vector V inside a view inside of the vector space V and some scalars C, the following properties are gonna be true. First of all, the zero vector is unique and we call it the zero vector. What that means is there's only one vector in the vector space that acts like the zero vector. Now sometimes we get a little ahead of ourselves when we think of the zero vector. Like when we think of the arrow interpretation, the zero vector is just a point. When we think of column vectors, the zero vector is just the array of all zeros. If we think of linear equations, which was an example we saw, we take the zero equation zero equals zero. Now we get so often used to thinking of that, there's just the one zero vector, but why is that? Could there be some foreign species of vector spaces that we find in like the Amazon Rainforest for which there are multiple vectors that act like the zero vector? And the answer to that is going to be no. The proof of that basically is the following. Let's prove, we're gonna use a technique which is called proof by contradiction. Suppose there are two zero vectors. There are two zero vectors. And we're gonna call the first one zero and we're gonna call the other one theta because theta is a Greek letter that kind of looks like zero. And so since there's zero vectors, what that means is the following. For any vector will take V inside of our vector space, will take V inside of our vector space. We have that zero plus V equals V and we get V plus zero also adds up to be V, right? If zero is a zero vector, then when you add it to V, you'll always get back V. But this is also true for the other vector theta that we said was a zero vector. If you add theta to V, you'll get back V. Or if you take, if you add V to theta, you'll get back V because if these are both zero vectors, then both of these properties have to happen. So then what you wanna do is you wanna sort of play the game. What happens if you take zero plus theta? Well, because zero is a zero vector, if you basically you're gonna take theta to be the other vector, if zero is a zero vector, then zero plus theta will equal theta, right? But on the other side, if theta is a zero vector, adding it to zero, you could replace V with zero right here. Adding theta to zero will give you back zero. And you can see then what happens is that theta would actually have to equal zero. So if you have two zero vectors, we can actually argue they actually were one and the same thing, thus proving that these things were unique, all right? So this is the first property of vector spaces. This property is true for every vector space that ever could happen. And so in fact, you could use this property to show that something's not a vector space. If a set of numbers has two zeros, then in fact, it's not a vector space anymore. What about additive inverses, right? The additive inverse of a vector is likewise unique. The same type of idea going on here. So let's say you be a vector inside our space and we're gonna let say negative u and u prime, these are gonna be vectors inside of our vector space such that, well, we want that u plus negative u, that should equal the zero vector. And it doesn't matter which order you go in, right? So negative u plus u is likewise equal to zero vector. But we also wanted to be true that if you add u with u inverse, this should also equal the zero vector. Because after all, this is what an inverse means. An inverse vector is a vector which when added to the original vector gives you always the zero vector, which we know is now unique. And so we can basically play the same game that we did earlier, right? So take u plus u prime plus u inverse, negative u, right? So consider this situation. What happens when you add u with u inverse and u prime, right? Well, because vector addition is associative, right? We can consider one of these things first. U plus u prime is equal to the zero vector. We're using the blue property there. So I'm gonna put it in blue. U plus u prime is equal to the zero vector. And when you add the zero vector at anything, you get back that vector. So this sum should equal negative u. But on the other hand, if we redid the parentheses, so we had negative u plus u plus u prime. Well, in that situation, this guy right here is gonna equal zero. It's the zero vector plus u prime, which then equals u prime here. And so then what we see again is that u prime has to equal negative u. And so there's only one inverse vector associated to any vector right here. So if a vector walks like an inverse and quacks like an inverse, then it must be the inverse. There's only one inverse vector possible because of this type of thing. And this is just a simple little argument using the axioms of the vector space, things that are true for every vector space. So it doesn't matter if we go to Pluto and we look up plutonian vector spaces, all of them will have unique inverses per vector. Another property that's kind of very interesting here is that property three says that if you scale any vector by u, you end up with the zero vector. It's times by zero always gives you the zero vector. And so the proof of that is the following type argument. What we're gonna do is we're gonna start off with u times zero times u. Now, because zero here, this is the scalar zero, zero has the property that it's zero plus zero. So you could replace zero with its sum of zero plus zero. Now, scalar multiplication distributes here in which case this will equal zero u plus zero u. And so what we've now shown is that zero u equals zero u plus zero u. Now, whatever that vector is zero u, it has an additive inverse. So what we can do is we can add negative zero u to both sides of the equation. We can add it over here as well. We can add negative zero u to both sides. Now, on the left-hand side, this would combine together to give us the zero vector. On the right-hand side, one of those will cancel out and we end up with zero u plus the zero vector. But adding the zero vector to something always just gives you back the other vector. So what we now see is that zero times u is equal to the zero vector. And so this gives us some proofs of these very useful properties right here. Now, the other one's property four tells us that if you take any scalar multiple of the zero vector, you get back the zero vector. This is gonna be left as an exercise for the student here, but it's very, very similar to the proof we just did a moment ago. And then finally, also property five here, the additive inverse of a vector is just timesing that vector by negative one. And you're gonna do a very similar, this one's also left as a proof to the viewer right now, that the proof of this is also very similar to property three right here, because we know additive inverses are unique. And so what we're gonna do is you're gonna take u plus negative one times u, use the properties, like the distributive property of a vector space, and then show that this has got to equal the zero vector. And so therefore, if u plus this guy is equal to the zero vector, since inverses are unique, it must be that it's equal to negative u, like that. And so these are some basic properties of vector spaces. Every vector space has these five properties. Therefore, if you ever find a set of objects that claims to be a vector space, but one of these five properties fails, that means it's not actually a vector space, and that actually provides a counter example to it not being a vector space. I make mention of this because there are some exercises in this section for which you are gonna be asked to play a thought experiment. That thought experiment would be, what if we change the definition of vector addition or scaling multiplication for column vectors? What if we changed it, so it wasn't the rules that we saw earlier, why would that be a bad idea? And so if we follow that thought experiment, what we're gonna see is that the new definition of vector addition or the new definition of scale multiplication will violate one of the eight axioms that we had previously. And one way to see the violation, it could be to find a counter example of the axiom itself, but it could also be to find a counter example of one of these properties right here, because if one of these properties is violated, it means one of the axioms is violated, therefore, you don't have a vector space. So you might wanna think about these properties if you're doing that exercise of the thought experiment of what if the vector space was different, right? And I want you to kind of pretend like you're a child at the moment, right? Why can't I be a train conductor to the moon, right? Maybe as a child, that's what you wanna do, you love the moon, you love trains, why can't I drive a train to the moon? Maybe you can, maybe you can't, right? Play around with that idea for a moment. You might not, you might be surprised what you discover as you allow yourself to think outside of the proverbial box right here, right? What if we think of our algebra in a different way? Why does it have to be the way that I told you? Why couldn't it be something different? Because sometimes the answer is it can't be something different because it doesn't work and you then figure out why it doesn't work. But sometimes your mind gets blown because when you think about it, why can't it be different? You actually discover that maybe it could, right? Your scalars don't have to be real numbers. Maybe they could be coming from some finite field like ZP or the complex field C. We learn a lot by allowing ourselves to push the boundaries when we ask the question, why? Why does it have to be this way? And the answer to that question, whether it does have to be that way or no, it doesn't have to be that way, the answer in either situation is illuminating, helps us understand the subject matter more. So try to be patient with yourself and think about these things. What would happen if I change the norm and follow it through to the end here? Speaking of following it through to the end, that actually does bring us to the end of section 1.3. So thanks for watching, everyone. If you feel like you learned something or you liked what you learned, give this video a like. Any of the videos you watch by all means, like them, like them, like them, right? Because they're amazing. Feel free to subscribe if you wanna see more updates and videos like this in the future. And hopefully I'll see you next time in section 1.4. Bye, everyone.