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Mathematics for Machine Learning and Data Science
Beginner
Topics
Deep Learning
Mathematical Foundations
Supervised Learning
As I mentioned before, equations behave a lot like sentences as they are statements that give you information. In this video, you will learn what a linear equation is, and what a system of linear equations is. As a matter of fact, you will be solving your first system of linear equations, which is extracting all the possible information from that system. Just like with systems of sentences, systems of linear equations can also be singular or non-singular based on how much information they carry. And as you already learned these concepts with real-life sentences, you are more than ready to tackle them with equations. In the previous video, you saw sentences such as, Between the dog and the cat, one is black. For the rest of the course, you'll focus on sentences that carry numerical information, such as this one. The price of an apple and a banana is $10. This sentence can easily be turned into equations as follows. If A is the price of an apple and B is the price of a banana, then the equation stemming from this sentence is A plus B equals 10. Of course, you might wonder why an apple and a banana together cost a whopping $10. To keep our example simple, I'm going to use these hypothetical prices with nice whole numbers. So while the thought of a $10 apple and banana might make your wallet shudder, rest assured, it's all in the name of mathematical simplicity. Now here's the first quiz in which you will be solving the first system of linear equations in this class. The problem is the following. You are going to a grocery store, but this is a very peculiar grocery store. In this store, the individual items don't have information about their prices. You only get the information about the total price when you pay in the register. Naturally, being a math person as you are, you are interested in figuring out the price of each item, so you keep track of the total prices of different combinations of items in order to deduce the individual prices. So the first day that you go to the store, you bought an apple and a banana and they cost $10. The second day, you bought an apple and two bananas and they cost $12. And the question is, how much does each fruit cost? So several things may happen. You may be able to figure out the price of the apple and banana, or you may conclude that you don't have enough information to figure this out. Or even more, you may conclude that there was a mistake with the prices given this information. All of these are options in the quiz. And the solution is that apples cost $8 and bananas cost $2 each. Why? Well, from day 1, you can see that an apple plus a banana is $10. From day 2, you can see that an apple plus two bananas is $12. So what was the difference between day 1 and day 2? Well in day 2, you bought one more banana than in day 1. Also, in day 2, you paid $2 more than in day 1. Thus, you can safely conclude that that extra banana you bought on day 2 cost $2. The extra $2 you paid on day 2 were because of that extra banana you bought on day 2. And now that you know that bananas cost $2, well how much do apples cost? Well, from day 1, you can see that an apple and a banana cost $10. So if a banana costs $2, then the remaining $8 must correspond to the apple. Thus, each apple costs $8 and each banana costs $2. To start, you'll be solving a first system of three equations with three unknowns. And the problem is a familiar one. You are in a similar store as before, but now your goal is to find the prices of three items. An apple, a banana, and a cherry. So you go to the store three days in a row. On the first day, you bought an apple, a banana, and a cherry and paid $10. On the second day, you bought an apple, two bananas, and a cherry and paid $15. And on the third day, you bought an apple, a banana, and two cherries and paid $12. Now the question is, how much does each fruit cost? So the way to solve this is very similar to the way to solve the previous systems with two equations and two unknowns. The first equation says that an apple, a banana, and a cherry cost $10. The second one says that the same arrangement plus an extra banana costs $15, therefore that extra banana must cost those extra $5. And the third one says that the same arrangement of day 1 plus an extra cherry costs $12, therefore that extra cherry must cost that extra $2. So banana is $5 and cherry is $2. You still need to find the price of the apple. Now look at the first equation. If the banana is $5 and the cherry is $2 and the three of them cost $10, then the apple must cost $3. So our solutions are the price of an apple is $3, the price of a banana is $5, and the price of a cherry is $2. The system of equations you just solved is this one over here, system of equations 1, which has as equations a plus b plus c equals 10, a plus 2b plus c equals 15, and a plus b plus 2c equals 12. And the solution you got was a equals 3, b equals 5, and c equals 2. Now here's quiz 2. The scenario is the same except the prices in the store are a little different. And you also bought different quantities of fruits. On day 1, you bought an apple and a banana and they cost $10. On day 2, you bought two apples and two bananas and they cost $20. The question is, how much does each fruit cost? Remember that the options of not having enough information or having a mistake in the information given are both valid as well. For this problem, the solution is that there is not enough information to tell the actual prices. And why is this? Well, you can use a similar reasoning than before. From day 1, you can deduce that an apple and a banana cost $10. From day 2, you can deduce that two apples and two bananas cost $20. Now these two equations are the same thing. They may not look the same, but in disguise, they're the exact same thing. Because you see, if one apple and one banana cost $10, then twice of one apple and one banana cost twice of $10, which is $20. So two apples and two bananas cost $20. Therefore, the system is redundant because it basically has the same equation twice. It's like that system of sentences where both sentences stated that the dog was black. This system didn't carry enough information. Now what are the solutions to this system? Well, because the system doesn't carry enough information, the system has infinitely many solutions. Any two numbers that add to 10 are a solution to this system. So for example, if the apple is 8 and the banana is 2, then that works because apple plus banana is 10 and 2 apples plus 2 bananas is 20. But if they're 5 and 5, that also works. If they're 8.3 and 1.7, that also works. And even saying that the apples are free and the bananas are 10 works too. So this system has infinitely many solutions because you simply don't have enough information. You don't have the two equations to narrow it down to one single solution like you had with the complete system. And now you're ready for a final quiz. Similar scenario except the first day you bought an apple and a banana and they cost $10 and the second day you bought two apples and two bananas and they cost $24. Can you figure out how much each fruit costs? And remember, there are still the options of not enough information or a mistake in the information. And the answer here is that there is no solution. Why? Well, in the same fashion as before, if one apple and one banana cost $10, then two apples and two bananas must cost $20. But the store charged you $24 for two apples and two bananas. Where are those four extra dollars? If you assume that there are no extra fees for buying more than one fruit or discounts or anything of that sort, then you must conclude that that extra money must be due to a mistake with the register when you checked out in at least one of the two days. This means that these two equations contradict each other. Just like the two sentences the dog is black and the dog is white contradicted themselves. And this concludes that the system has no solutions. So here's a recap. You solved three systems of equations. The first one has the equations A plus B equals 10 and A plus 2B equals 12, because the price of an apple and a banana was 10 and the price of an apple and two bananas were 12. The second one has the equations A plus B equals 10 and 2A plus 2B equals 20. And the third one has the equations A plus B equals 10 and 2A plus 2B equals 24. The first one had a unique solution, which was A equals 8 and B equals 2, for A is the price of an apple and B the price of a banana. The reason this system has a unique solution is because both equations give you one different piece of information, thus you're able to narrow down the solution to one unique solution. For this reason the system is complete and non-singular. The second system has infinitely many solutions, which are any two numbers that add to 10. In this system, the two equations are the exact same one, so you never had a second equation to help you narrow down the solution to a unique one. This means the system is redundant and singular. And finally, the third system has no solution because the two equations contradict each other. Therefore, this system of equations is contradictory and singular. So as you see, we're using the same terminology as with systems of sentences and everything works in the exact same way. Now you're ready to solve more systems. Here are three systems to solve. Remember that just like before, some of them may not have a solution, some of them may have an infinite number of solutions. So if that's the case, please state it, even if any of the options already shows a solution to the system. And so here are the solutions. System 2 has infinitely many solutions. Why? Well, you look at equation 1 and it says A plus B plus C equals 10 and from equation 2, there's an extra C and an extra 5. So that C must be equal to 5. The price of a cherry is 5. However, when you go from equation 2 to 3, there's also an extra 5 and also an extra C. So equation 3 brings nothing new to the table. When you replace C equals 5 on the first equation, you get A plus B equals 5. And actually, any triplet where the third number is 5 and the first 2 add to 5 works. So all of these are solutions to that system. That system has infinitely many solutions. Now let's look at system 3. System 3 has no solutions. Why? Well, from the first and the second equation, C is equal to 5. However, from the second and the third, C is equal to 3 because from the second equation to the third, when you bought an extra cherry and you paid an extra $3. So C is equal to 5, but C is equal to 3. That's a contradiction. So system 3 has no solution. What about system 4? Well, system 4 has infinitely many solutions. Why? Because as you can see, the second equation is 2 times the first one, and the third equation is 3 times the first one. So the first one is really the only equation here that matters. Or equivalently, the second or the third, but only one of them matters. So any three numbers that add to 10 would work. So for example, 0, 0, 10, 2, 7, 1, 9, 1, 0. Any three numbers that add to 10 are a solution, so we have infinitely many solutions. And finally, some clarification. You may have noticed the word linear equation several times. What does that mean? Well, linear equation can be anything like a plus b equals 10, 2a plus 3b equals 15, 3.4a minus 48.99b plus 2c equals 122.5, anything like that. And notice that it can have as many variables as we want, but there's a special rule that must be followed. In a linear equation, variables a, b, c, etc. We're only allowed to have numbers or scalars attached to them. And there's also an extra number all by itself, like the 122.5 here allowed to be in the equation. So in short, you can multiply the variable by scalars and then add them or subtract them and then add a constant. And that's it. So what's an equation that's nonlinear? Well, nonlinear equations can be much more complicated. They can have squares like a squared, b squared. They can have things like sine, cosine, tangent, arc tan, anything like that, powers like b to the 5. They can have powers like 2 to the a or 3 to the b. And furthermore, you can actually multiply the a's and b's. In linear equations, you can only add them, but in a nonlinear equation, you can have a, b, a, b squared, b divided by a, 3 divided by b, things like logarithms, anything along those lines. So linear algebra is the study of linear equations like the ones in the left. And since they're much simpler, then there are many things you can do with them, such as manipulating them and extracting information out of them. So we're only going to worry about the linear equations in the left and the reasons called linear algebras because it's the study of linear equations.
