 Welcome back everyone. We're continuing our discussion in section 1.3 of our textbook about vectors and vector spaces. In the previous video, we learned about what a vector space is and we saw the canonical example of a vector space that is these physical arrows in space that could represent velocity, it could represent force, just as some examples we would see in physics, right? But our vectors are literally arrows for which we can operate addition and scale multiplication using things like the parallelogram rule and such. Now, another type of vector is actually just an array of numbers. So in this context, our vectors are going to be arrays, arrays of numbers. Now I have to be careful what I mean by number here because we have scalar numbers, we have vector numbers. So these are gonna be arrays of scalars which are numbers which come out of a field. And typically these arrays are gonna be written vertically. So for example, we might take something like this array, one, one, two, three. So we'll write this as an array where we write the first number on top, then the next number, then the next number and however many numbers there happens to be in the array. Now this is how we're most commonly gonna denote our vectors but sometimes it makes more sense from like a type setting point of view, so in this case our vector could be written vertically but we could also write it horizontally. We might say something like one, two, three. And in this context, we make no distinction between these things because the vector itself is an array of numbers. We have to know the first number, the second number, the third number. The array is ordered right. If we switch up the order a little bit, this is not considered the same vector. There's a first number, there's a second number, there's a third number and that matters. That matters a lot, right? And also the length of the, like the number of components in the vector also matters, right? If you had something like one, one, two, three, this would be considered a different vector still because this vector has four components, this vector has two, or sorry, three components and even though it's the same three numbers, one, two and three, the fact that one is repeated here but not here distinguishes between these vectors. And so the vector itself is this array of numbers that the order matters, the number of numbers matters but the exact arrangement we really don't care about. We could write it vertically, we could write it horizontally and that's fine. Now, one thing I do wanna caution you about is that if you want to write your vector horizontally, do not write it with brackets like the following, right? We're not gonna do that. Now you might be like, oh, you did brackets over here, why does that matter? Well, then the thing is at this context is we're learning about vectors here. This object right here is referred to as a column, oftentimes called a column vector to distinguish it from other types of vectors one could study and then then actually this one right here, we would call a row vector. And now as we think about vectors, there's no distinction between writing things vertically or horizontally but later on, particularly in chapter three and beyond, when we start focusing more on matrices, which matrices are gonna be two dimensional arrays of numbers as opposed to these vectors, which are one dimensional. And in the two dimensional case, we actually do wanna distinguish between vertical and horizontal, that is, we wanna distinguish between column vectors and row vectors. And so because of that, we don't wanna get into the habit of doing that right now. And so when you write your, if you write your vectors horizontally, avoid using the square brackets. Instead, if you want to write things horizontally because you feel like it works better on the page, put parenthesis, put parentheses around it, much like we would write a point in space because really we're not making any distinction between points in space and these vectors here. Sometimes linear algebra textbook, calculus textbook, try to put this huge emphasis on that vectors, vectors and points in space are two different things. But in fact, that's actually a huge misnomer. What we're saying here is that, oh, vectors are points in space because we can add them and we can scale them. What does it mean to add vectors in this context? If you have two vectors, column vectors here, so we have like some X1, some X2, and let's say there's n entries, and you wanna add this to some other vector, say Y1, Y2, up to Yn, how does one add these two vectors together? Well, what we do is we simply just add their components, X1 plus Y1, X2 plus Y2, all the way down to Xn plus Yn. We can add together the vectors just by adding together their components. And how does one scale a vector? Like if you have some scalar C and you times it by the vector X1, X2, all the way down to Xn. Well, scalar multiplication we defined is just times each of the components in the vector by the scalar C. So the scalar product would look like CX1, CX2, all the way down to CXn. And that's how we define the vector operations for these column vectors here. And this can be done for any field whatsoever. And so we actually introduced the notation F to the n right here. This is gonna equal the set of vectors. I should say the set of column vectors, column vectors with n entries. So there's n numbers in there. And I should also mention entries from the field F. And so some examples of this type of thing, we could talk about the vector space Rn. We could talk about the vector space Q3. So this would be vectors with three rational entries. We could talk about the complex vector space C2. This would be vectors with two numbers, which are complex numbers. And so this turns out to not be any more complicated than one might think. Let's do an example in R3. That is we take vectors in R3. What this means is that we'll take vectors with three numbers in the array and these numbers are gonna be real numbers. So case in point, take the vectors u and v here. So u is the vector containing six, negative two and two. v will be the vector who contains negative three, zero and five. And so when you add together u and v, well here's u, here's v. And so vector addition here just means add together the points in the same position. So you get six plus negative three or six minus three, which is three. For the second component, you'll take negative two plus zero as you see right here. That's just a negative two. And then lastly, for the third component, you'll take three, for the third component, you take two plus five, like you see here, which is just a seven. And so the vector sum of u plus v is just three, negative two, seven. It couldn't be simpler than that. The only difficulty here is the difficulty of arithmetic for the associated field. Scale and multiplication, same basic idea. Take the vector this time, we're gonna take v to be negative one, three and two and let's scale the vector by three. And if we do that, so three times this vector, that means we're gonna take three times negative one, which is negative three. We take three times three, which is a nine and we take three times two, which is equal to six. And that's all there is to this, to this scalar multiplication and vector addition. And one could show that these simple operations of vector addition and scalar multiplication give us the axioms of the vectors, make those eight properties we saw previously. And in fact, essentially, these are the only ways we can define vector addition and scale multiplication for column vectors and to guarantee those axioms, you'll explore that principle in the homework a little bit.