writing linear systems down can get a little bit tedious particularly when you have a whole bunch of different equations that a whole bunch of different variables it kind of cumbersome to write down fold them so what we're going to do in this video is introduce a new notation called matrix notation it's just going to make ourselves a little bit more efficient for writing down linear systems it's gonna be something we're gonna do all the time and it's gonna be greatly appreciate it so we have this little bit of efficiency first if I look at what
I have here I have a lot of repetition notice that there's an X 1 and this this X 1 appears every single time in this first column and then there's an X 2 that appears every single time in the second column sometimes I kind of hide it if you'll notice on the second equation I don't have it written explicitly but I can fill it in as plus 0 times X 2 and I can do it down here at the bottom or this is plus 0 times X 3 so whether I write it explicitly or not
there's a whole column of X 1 with coefficients the whole comb of X 2's with coefficients and a whole column of X trees with coefficients alright so how would I just don't write down the X 1 or the X 2 what I'm going to do with the matrix and our notation is just a big set of brackets here and then in each of my columns I'm not going to write the coefficient and the variable I'm just going to write the coefficient so the first column it was 1 times X 1 so that's a 1 1
times X 1 again so that's the 1 7 times X 1 that's 7 and then 2 times X 1 so that's a 2 in the second column it's a 3 times an X 2 I already filled it in so this is a 0 times an X to a minus 2 times the next 2 and a 0 a minus 1 a 1 a 0 and a 2 and finally on the far right-hand after the equal signs 1 2 7 4 and it try to keep the equal signs in there I might come along and put this
sort of dotted line here that sort of separates it so the stuff on the left hand side is the thing that's variable the stuff on the right hand side is just going to be those constants a few little pieces of terminology here this portion the portion of the maker set to the left of my dotted line here that refers to all of the coefficients to the variable this is uninspiring ly referred to as the coefficient matrix the portion which is on the right hand side the stuff that corresponds to the constants is referred to as
the constant matrix sometimes the constant vector we'll see what that is in a little while but for now constant matrix and then the entire thing consisting of both the coefficients and the constant is going to be the Augmented matrix so the entire thing is the Augmented matrix in sort of divides up into these two different portions the coefficient matrix and the constant matrix and this is going to be how we're going to write down all of our linear systems in the future
