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Matrix word problem: vector combination

Video transcript
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 2 by 2. And in the future, we'll do 3 by 3's. We won't do 4 by 4's because those 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're more likely to see in your linear algebra class when you take it in college. But the really neat thing here is, and I think this will really hit the point home, that the matrix representation is just one way of representing multiple types of problems. And what's really cool is that if different problems can be represented the same way, it kind of tells you that they're the same problem. And that's called an isomorphism in math. That if you can reduce one problem into another problem, then all the work you did with one of them applies to the other. But anyway, let's figure out a new way that 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. And I'm going to just write this is as a column vector. And all of this is just convention. These are just human invented things. I could have written this diagonally. I could have written this however. But if I say vector a is 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 is equal to 2, 6. And I want to know are there some combinations of vectors a and b-- where you can say, 5 times vector a, plus 3 times vector b, or 10 times victor a minus 6 times vector b-- some combination of vector a and b, where I can get vector c. And vector c is the vector 7, 6. So let me see if I can visually draw this problem. So let me draw the coordinate axes. Let's see this one. 3, negative 6. That'll be in quadrant-- these are both in the first quadrant. So I just want to figure out how much of the axes I need to draw. So let's see-- Let me do a different color. That's my y-axis. I'm not drawing the second or third quadrants, because I don't think our vectors show up there. And then this is the x-axis. Let me draw each of these vectors. So first I'll do vector a. That's 3, negative 6. 1, 2, 3, and then negative 6. 1, 2, 3, 4, 5, 6. So it's there. So if I wanted to draw it as a vector, usually start at the origin. And it doesn't have to start at the origin like that. I'm just choosing to. You can move around a vector. It just has to have the same orientation and the same magnitude. So that is vector a for the green. Now let me do in magenta, I'll do vector b. That is 2, 6. 1, 2, 3, 4, 5, 6. So 2, 6 is right over there. And that's vector b. So it'll look like this. That's vector b. And then let me write 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 7, 6. Let me do that in purple. So 1, 2, 3, 4, 5, 6, 7. Comma 6. So 7, 6 is right over there. That's vector c. Vector c looks like that. I'm going to draw it like that. And that's vector c. So what was 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 multiply times vector a is x. And the multiple of vector b is y. So I essentially want to say that-- let me do it in another neutral color-- that vector ax-- that's how much of vector a I'm contributing-- plus vector by-- 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 expand out vectors a and b. Vector a is what? 3, negative 6. So vector a, we could write as 3, minus 6 times x. That just tells us how much vector a we're contributing. Plus vector b, which is 2, 6. And then y is how much vector b we're contributing. And that is equal to 7, 6. Vector c. Now this right here, this problem can be rewritten just based on how we've defined matrix multiplication, et cetera, et cetera, as this. As 3, minus 6, 2, 6, times x, y, is equal to 7, 6. Now how does that work out? Well think about how matrix multiplication works out. The way we learned matrix multiplication, we said, 3 times x, plus 2 times y is equal to 7. 3 times x plus 2 times y is equal to 7. That's how we learned matrix multiplication. That's the same thing here. 3 times x, plus 2 times y, is going to be equal to 7. These x and y here are just scalar numbers. So 3 times x plus 2 times y is equal to 7. And then matrix multiplication here, minus 6 times x plus 6 times y is equal to 6. That's just traditional matrix multiplication that we learned several videos ago. That's the same thing here. Minus 6x plus 6y is equal to 6. These x's and y's are just numbers. They're just scalar numbers. They're not vectors or anything. We would 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 a-ha moment now, if you watched the previous video. Because this matrix also represented the problem, where do we find the intersection of two lines? Where the two lines-- I'm just going to do it on the side here-- the intersection of the two lines, 3x plus 2y is equal to 7. And minus 6x plus 6y is equal to 6. And so, I had drawn two lines. And we said, what's the point of intersection, et cetera, et cetera. And it was represented by this problem. But here, we have-- well 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 trying 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 the same exactly way we solved this problem. If we call this the matrix a, let's figure out a inverse. So we get 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 1/30. And we did this in the previous video. You swap these two numbers. So you get 6 and 3. And then you make these two 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, this cancels out. So you get x, y is equal to a inverse times this. It's equal to 1/30 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 did it here. If we did it the other way, all bets are off. So what does this equal? This is equal to 1/30 times-- and we did this the previous problem-- 6 times 7 is 42, minus 12. 30. 6 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. 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 let's see, vector b is over 2 and up 6. So over 2 and up 6 would get us there. So 1, vector b-- just doing heads to tail visual method of adding vectors-- would get us there. 1, 2, 3. Good. No, let me see. 1, 2, 3. And then vector b goes over two more. two more. So it'll get us up 6. It's like that. So that's 1, vector b. And then if we add another-- but we want 2 times vector b. We essentially need two vector b's. So we had one, and then we add another one. I think visually you see that it does actually-- I didn't want to do it like that. I wanted to use the line tool so it looks neat. So you add another vector b. And there you have it. That's a vector b. So it's 2 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 algabraically. But the real learning, and the big real discovery of this whole video, is to show you that the matrix representation can represent multiple different problems. This was a finding the combinations of a vector problem. And the previous one it was figure out if two lines can intersect. But what it tells you is that these two problems are connected in some deep way. That if we take the veneer of reality, that underlying it, they are the same thing. And frankly, 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 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 is tying all of these different concepts together. But anyway, I don't want to get all mystical on you. But 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-- A lot of people take linear algebra, they learn how to do all of the things, and they say, well what is the whole point of this? But this is kind of an interesting thing to think about. We had this had 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 it up, we got vector c. So an interesting question is, what are all the vectors that I could get to by adding combinations of vectors a and b. Or adding or subtracting. Or you could say, 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 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've had n dimensional vectors. So it gets really, really, really abstract. But this is, I think, a really good toe dipping for linear algebra as well. So hopefully I haven't confused or overwhelmed you. And I'll see you in the next video.