Matrix word problem: vector combination

in the last video we saw how a matrix and figuring out its inverse can be used to solve a system of equations and we did a 2x2 and in the future we'll do three by threes when won't do four by fours because let's take too long but you'll see it applies to kind of an N by n matrix and that is probably the application of matrices that you learn if you learn this in your algebra 2 class or your algebra 1 class and you often wonder well why even do the whole matrix thing now I will show you another application of matrices that actually you'll you're more likely to see in your linear algebra class when you take it in college but but the really neat thing here is to show that I think this will really hit the point home that the matrix representation is just one way of representing multiple types of problems that what's really cool is that if different problems can be represented the same way it kind of tells you that the same problem that's called an isomorphism in math that if you can reduce one problem into another problem then all of the work you did with one of them applies to the other but anyway let's so let's figure out a new way that matrix matrices can be used so I'm going to draw some vectors let's say I have the vector let's call this vector a let's say I have vector a and it equals and I'm going to just write this as a column vector and in all of this is just convention these are just human invented things I could have written this diagonally I could written this however but if I say vector a is 3 comma negative 6 not comma but you know 3 negative 6 and I view this as the X component of the vector and this is equal to the Y component of the vector and then I have vector B vector B vector B is equal to vector B I have Z equal to let's move me make sure I get this 2 comma 6 and I want to know I want to know is there some combinations of vector a and B where you know I could say you know I don't know five times vector a plus three times vector B or 10 times vector a minus six going to be some combination of vector a and B where I can get vector C where I can get vector C and vector C is the vector is the vector seven six so let me see if I can visually draw this problem so let me draw the coordinate axes let's see this one three comma negative 6 that'll be in court all these are both in the first chord so I just want to figure out what I how much I need to how much of the axes I need to draw so let's see so if this is the Y let me do a different color I do a different color that's my y-axis I'm not doing drawing the second or third quadrants because I don't think our vector is show up there and then this is this is the x-axis let me draw each of these vectors so first I'll do vector a vector a that's 3 comma negative 6 so 1 2 3 and the negative 6 1 2 3 4 5 6 so it's there so if I wanted to draw it as a vector visually start at the origin and it doesn't have to start at the origin like that I'm just choosing to you can you can move around a vector it just has to have the same orientation and the same magnitude alright so that is vector a that's vector a for the green now let me do in magenta I'll do vector B that is 2 comma 6 1 2 3 4 5 6 so 2 comma 6 so 2 comma 6 is right over there and that's a vector B vector B so it'll look like this let's vector B and let me do vector right down vector a down there that's vector a and I want to take some combination of vectors a and B and add them up and get vector C so what does vector C look like it's seven comma six let me do that in purple so one two three four five six seven comma six so seven comma six is right over there that's vector C vector C is looks like that vector C let me oh yeah I want to draw it like that and that's vector C so what is the original problem I said I said I want to add some multiple of vector a to some multiple of vector B and get vector C and I want to see what those multiples are so let's say the multiple that I multiplied times vector a is X and the multiple of vector B is y so I essentially want to say that I'm going to do it in another neutral color that vector a X right that's how much of vector a I'm contributing plus vector B Y that's how much of vector B I'm contributing is equal to vector C and you know maybe I can't maybe there's no combinations of vector a and B when you add them together equal vector C but let's see if we can solve this so how do we solve so let's let's expand out vectors a and B a vector a is what three comma negative six so this is vector a we could write it as 3 minus 6 times X right that just tells us how much vector a we're contributing Plus vector B which is 2 comma 6 which is 2 comma 6 and then Y is how much of vector B we're contributing and that is equal to seven comma six vector-c now this right here this problem can be rewritten just based on how we've defined matrix multiplication etc etc as this as three - six - six times X Y is equal to seven six now how does that work out well think about how matrix multiplication works out the way we learn matrix multiplication we said oh you know 3 3 times 3 times X plus 2 times y is equal to 7 right 3 times X plus 2 times y is equal to 7 that's how we learn matrix multiplication but that's the same thing here 3 times X plus 2 times y is going to be equal to 7 right these x and y here or just scalar numbers right so 3 times X plus 2 times y is equal 7 and then matrix multiplication here - 6 times X plus 6 times y is equal to 6 that's just traditional matrix multiplication that we learn several videos ago that's the same thing here - 6 X + 6 y is equal to 6 these are just these X's and Y's or just numbers they're just scalar numbers and not vectors or anything so we just you just multiply them times both of these numbers so hopefully you see that this problem is the exact same thing as this problem and you've maybe had an aha moment now if you've watched the previous video because this matrix this matrix also represented the problem well where do we find the intersection of two lines where the two lines and I'm going to do it on the side here you know the intersection of the two lines 3x plus 2y is equal to 7 and minus 6x plus 6y is equal to 6 and so you know I had drawn two lines and we said oh what's the point of intersection X etc and it was represented by this problem but here we have but I won't say a completely different problem because we're learning they're actually very similar but here I'm doing a problem of I'm to find what combination of the matrices a and B add up to the matrix C but it got reduced to the same matrix representation and so we can solve this with the same exact way we solved this problem if we call this the matrix a let's figure out a inverse so we get a inverse let me say a inverse is equal to what it equals 1 over the determinant of a the determinant of a is 3 times 6 18 minus minus 12 so that's 18 plus 12 which is 1 over 30 and we did this in the previous video you swap these two numbers so you get 6 and 3 and then you make these 2 negatives so you get 6 and minus 2 that's a inverse and now to solve for x and y we can multiply both sides of this equation by a inverse if you multiply a inverse times a it can't this cancels out so you get you get X comma y is equal to a inverse times this so it's equal to 1 over 30 times 6 minus 2 6 3 times 7 6 and remember with matrices the order that you multiply matters so on this side we multiplied a inverse on this side of the equation so we have to do a inverse on the left side on this side of this equation so that's why I did it here if we did it the other way all bets are off so what is this equal this is equal to 1 over 30 times and we did this in the previous problem 6 times 7 is 42 minus 12 36 times 7 42 plus 18 60 so that equals 1/2 so what does this tell us this tells us that if we have 1 times vector A plus 2 times vector B right 1 times this is 1 and 2 times vector B so 1 times vector A plus 2 times vector B is equal to vector C and let's confirm that visually so 1 times vector a well that's vector a right there so if we add 2 vector B's to it we should get vector C so let's see if we can do that so if we just shift vector B over this way well vector B let's see vector B's over two and up six so over two and up six would get us there so one vector B just doing heads to tail visual method of adding vectors would get us there right one two good I think now let me see it's it let's see one two three and then vector B it goes over two more two more and then so it'll get us an up six looks like that so that's one vector B and then if we add another but we want to times vector B right we essentially have made two vector B so we add one and then we add another one I think visually you see that it does actually well I didn't want to do it like that I wanted to use the line tool so it looks neat so yet another vector B and there you have it that's a vector B so it's two times vector B so it's the same direction as vector B but it's two times the length so we visually showed it we solved it algebraically but the real learning and the big real discovery of this whole video is to show you that this matrix this the matrix representation can represent multiple different problems this was a at you know finding the combinations of vector problem and in the previous one it was figure out if two lines can intersect but but what it tells you something is that these two problems are connected in some deep way that you know if we take the veneer of reality that underlying it they are the same thing and you know frankly that that's why math is so interesting because when you realize that two problems are really the same thing it takes all of the superficial human veneer away from things because you know our brains are kind of wired to perceive the world in a certain way but it tells us that there's some fundamental truth independent of our perception that that is tying all of these different concepts together but anyway I don't want to get all mystical on you but if if you do see the mysticism in math all the better but hopefully you found that pretty interesting and actually I know I'm going over time but I think this is kind of you know a lot of people they take linear algebra they learn how to do all of the cool things and they said what was the whole point of this but this is kind of an interesting thing to think about you know we had this matrix a we have this that we start we have this vector a and we had this vector B and we were able to say well there's some combinations of the vectors a and B that when we added up we got vector C right so an interesting question is what are all the vectors that I could that I could get to by adding combinations of vectors a and B or adding or subtracting or or you could say you know I could multiply them by negative numbers but either way what are all of the vectors that I can get by taking linear combinations of vectors a and B and that's actually called the vector space spanned by the vectors a and B and we'll do the more of that in linear algebra and here we're dealing with a two dimensional Euclidean space we could have had three dimensional vectors we could have had n dimensional vectors so it gets really really really abstract but this is I think a really good toe tipping for for linear algebra as well well hopefully I haven't confused or overwhelmed you and I will see you in the next video