 Now we're ready to define the primary term for this section here. What is a linear combination? So we've defined now what a vector means, right? These are things that belong to a vector space and vectors are essentially these things that we can add and scale according to the standard laws of a vector space So now a linear combination to define that take us take some list of vectors here So v1 v2 up to vp This is p many vectors from some some vector space And take that some same number of scalars c1 c2 up to cp these are scalars that belong to the field and our vectors are belonging to an f vector space and then define some new vector y as the following Expression right y is equal to c1 v1. So we take the scalar product there We add that to and this would be a vector sum here We add that to c2 plus v2 add that to c3 v3 all the way down to cp vp and so in this situation y is then defined to be a linear combination of The vectors v1 v2 up to vp with Coefficient c1 c2 up to cp. So a linear combination is any any expression of vectors which combines together these additions and Scalar products of these things right now It's very possible, of course that the scalars here could just be one in which case any sum of Vectors is in fact a linear combination It could also be that p itself is the number one that is there's only one vector in the combination So a scalar multiple of the vector is a linear combination And so a linear combination is essentially the generalization of vector sums and scalar products That is as linear combination is all of the algebraic ways We could combine vectors together Using scalar multiplication and vector addition And so we're looking for all the ways we can combine vectors together and those are each examples of linear combinations Now if you just look at two examples here This first one let's compute a linear combination for a for a vector in c2, right? So this is the complex vector space with two entries So our vectors will look like column vectors with two components using complex numbers here So in this linear combination, we have two times the vector 3i and we have i times the vector 2i and 3 plus 5i now remember with working with complex numbers It's going to be necessary to remember that i is the square root of negative 1 And particularly what you need to know is that i squared is equal to negative 1 if you know that you're going to be fine here so when it comes to Linear combinations they follow the same order of operation as you're used to from algebra Multiplication takes precedence over addition. So we're going to first do the scalar products here So two times the vector 3i will give us six 2i for the first vector And for the second vector when we just when we scale by i you're going to get i times two Which is a 2i, but then you're going to get i times i which is negative one So I'm going to put the real number first For the next one you're going to get i times three which is a 3i and then you're going to get i times 5i Which is a negative five like so so we did the scalar product now we add together The numbers so we're going to add together the first component and we're going to add together the second component And these are complex numbers So add them together According to like terms the real parts go together and the imaginary parts go together So six minus one is a five and then zero i plus 2i is just a 2i the first number didn't have an imaginary part For the second component, there's no real part on the first one So you're just going to get a negative five there and then for the imaginary part you get 2i plus 3i Which is a 5i and so that gives us the linear combination of the numbers right there nothing nothing too complicated there Then As another example, this one here I forgot to mention this is going to be a Linear combination over the vector space z three three what that means here Is remember you might want to rewrite this thing a little bit. This is going to be z three I guess I should write my z like this z three cube, right? So what this right here means you're working mod three So our integers will reduce mod three all numbers should be either zero one or two And then the three up here as a superscript just means our vectors will have three components in the array So when you look at this right here, we're just adding together We'll take the linear combination zero one two plus two times one one one plus two times one zero two Uh, so be aware that we're working mod three in this situation Just like the last example, we're going to we're going to do all the scalar products first So we get zero one and two We're going to get two two two And then lastly we're going to get Two zero four now as we're working mod Three there is no such number as four exactly. It should be reduced down to be one And so remember when you work mod three two times two is actually equal to one Because four and one are congruent Modulo three So we do our scalar multiplication that changed that that did change a little bit, right? So instead of working over rational numbers working mod three, there was that reduction right here Now we add these numbers together Which in case you're going to get zero plus two plus two You're going to get one plus two plus zero and you're going to get two plus two plus one Like so if we just add these together as this integers we end up with four Three and five But remember as we're working mod three we want to reduce all of these things down And so the end four reduces to one like we saw Three would reduce to zero and five would reduce to two And so this linear combination would turn out to be the following vector one zero two I do want to mention that as Students are working through exercises if you're checking your answer with like the back of the book or anything like that Be aware that when you're working over a finite field such as z three all the vectors will be reduced modulo Whatever whatever the module says so if you didn't do that your answer might look different from the answer you're comparing it to It's not that you're wrong It's just you need to reduce it and therefore you need to recognize your answer might be Equivalence to the one presented in the back of the book there And that's how one can compute a linear combination It's no more difficult than adding vectors and scaling them based upon the arithmetic of the associated field