 Welcome back to our lecture series linear algebra done openly. As usual, I'm your professor today, Dr. Andrew Missildine. This video is the first video for section 2.1 about vector equations, which is the first section for chapter two, the algebra and geometry of vectors. We have seen before that vectors are simply quantities, which we can add and scale by the usual rules of associativity, commutivity, and the distributive laws. We've seen a lot of examples of diverse kinds of vectors, such as arrows. How do you add two arrows together? Well, in physics, there's sense to that meaning. We've seen linear equations could be viewed as vectors, polynomials could be viewed as vectors, just to name a few examples. A particular importance is the vector space fn. This fn, we keep on seeing over and over again like say rn, cn, c2, just as some examples here. What's so special about this fn? Well, f is just some field and n is a natural number, but recall this space, this fn, I'll put it back on the screen there, this fn is the set of column vectors, or row vectors if you prefer, but it's the set of column vectors whose entries come from the field f and there are n numbers in the array. We've seen examples of linear combinations of column vectors, linear transformations involving column vectors. In fact, every solution to a linear system that we have seen in this series thus far is in fact a column vector, although again sometimes it's written as a row. So, although there are all of these different types of vectors, there's variety and diversity to what we could consider a vector, why do we place so much focus primarily on column vectors? Well, the short answer to this question is that column vectors of fn are the most important type of vector. In this chapter, we're going to explore the topics of coordinates, which will ultimately prove that every finite dimensional vector space is isomorphic fn. Whoa, there might have been a lot of words I just used right there that you're not exactly familiar with yet. That's really okay. We'll be introduced to those topics more in this chapter. Basically, what I want you to take away from this discussion so far is that every vector space is essentially just fn. Maybe it's just enough disguise for Halloween or whatever. So, we're going to start learning more about vectors and see that they all essentially are just like fn, but that's near the end of this chapter. In the current section, 2.1, I want to talk some more about linear combinations. We've seen the linear combinations are really the bread and butter of vectors. We add vectors together. We scale them. And so, if you want to combine a bunch of vectors together to make a new vector, it's a linear combination. I just gave you a bunch of vectors, a1, a2, a3, up to an, and I gave you a bunch of scalars, x1, x2, x3, up to xn. You could combine those together by rules of the vector space, and that would then combine to be some vector, call it b, and we could then compute what b is in that situation the other direction. What if we are given the vectors a1, a2, up to an, they're just the same. And instead of being given the scalars, what if we're giving the vector b that they combined into? That is, what if we're given the following equation, like so, where the vectors a1, a2, up to an, they're given to us, they're fixed, and then the vector b is also given to us and fixed, but the scalars, x1, x2, these are variables. We don't know what they are. Could we figure out what the scalars are that combine the vectors a to give us the vector b? That's the question we have at hand, and so this is what we, by the vector equation. How does one solve this vector equation? Now to make life a little bit easier for us, each of these AIs, so like of a1, a2, a3, we'll actually, we'll call it aj right here, j will just be sort of like this variable to keep track of where we are in the list, you know, a1, a2, aj generically. So let's just say that each vector aj can be expressed in the following way. We'll call the first entry of aj, a1j, the second entry, a2j, the next entry will be a3j, all up to amj. So our vectors aj, we're going to say that these aj vectors belong to the vector space fm, m is the number of components there. Now the reason that because I don't claim that the number of entries in the vectors is the same as the number of vectors we have, those could potentially, those could potentially be different, right? So aj has m many elements in it and likewise the vector b will be a vector in fm and we'll call its entries just bj generically speaking. So with that, with that sort of like a notation I'm introducing to express the entries of these vectors, we see the following type of situation, vector equation x1 times a1 plus x2 times a2 all the way up to xn times an is equal to b. Remember the note that the convention we use here, we're going to write vectors in bold font and then scalars won't have any bold font so it's easier to tell them apart here. Okay? So if we were to expand what a1, a2, an are just by writing the numbers inside of them, remember a1 would look like a11, a21, a31 all the way down to am1, a2 would look like a12, a22, a32 all the way up to am2 and then if we go to the end of the list this will look like a1n, a2n, a3n all the way up to amn and then if we expand the b we get a b1, b2, bm. This is just writing the vectors as the column vectors they are. But now when you look at this right here we have scalar multiplication, you have a column vector times a scalar and the idea is you distribute this scalar onto each entry in the vector giving us what we see right here. This a11, x1, a21, x1, a31, x1 all the way up to am1, x1 and if we do this for every single vector along the way we'll get each of these. I guess we don't distribute anything on b. We didn't scale b by anything. So then we get each of these vectors if we do the scalar multiplication. But now we have these end vectors that we could add together. So what does it mean to add together column vectors? Well you add together the first components you just add those together you get this thing right here and then for the next one right here you add all the second components together which gives you this entry right here and then for the third components and the fourth components all the way down to the mth component you just add these all together and you get a combination like this. Now when you have two vectors that are equal to each other the only way that two vectors are equal to each other is if component wise they agree right the first components agree the second components agree the third components agree the mth components agree that's what vector equality means. So if we equate the first two entries together we get the following linear equation. We see that a11x plus a12x2 plus a13x3 plus a14x4 all the way down to a1nxn is going to equal b1. And we do that for the second entry which gives us the second equation. We do that for the third entry the fourth entry the fifth entry all the way down to the mth entry. And so now you can see why we chose the the numbering system that we did each of the entries in the vectors corresponds to the entry in a system of equations which will then translate to an augmented matrix right for which that's how we usually denote the entries of a matrix is the one one position the one two position the one three position the one n position the one two sorry the two one position the two two position the two three the m1 the m2 the m3 the mn. So you can see we we kind of knew what was going to happen here solving that vector equation corresponds to the following linear system of equations solving the vector equation requires us we solve the linear system write the linear system as an augmented matrix and then we reduce it to echelon form using gaussian or grunjauss gauss elimination. And so summarizing what we see right here is the following the solution solution to the any so so we have this solution vector right here the solution to the vector equation is likewise the solution to linear system which comes from the corresponding system of the corresponding matrix equation there this should be an f in there but despite the slight typo the solving the vector equation is equivalent to solving a system of equations for which augmented matrices is a very powerful tool in the next video we'll do some examples of solving vector equations by recognizing the corresponding linear system and then solving said linear system