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Let's take a look at a detailed example that uses the new array programming paradigm. We'll have NumPy do some linear algebra for us!

#### Code

```
orders = np.array([
[2, 0, 0, 0],
[4, 1, 2, 2],
[0, 1, 0, 1],
[6, 0, 1, 2]
])
```

```
totals = np.array([3, 20.50, 10, 14.25])
```

#### Learn More

- Build a Learning Mindset
- Wikipedia - Array Programming
- Khan Academy - Linear Algebra <== Wonderful courses
- NumPy also has a
`matrix`

datatype. This is for historical reasons. Learn more about`array`

vs.`matrix`

#### My Notes from Manipulation

```
## Array Manipulation
* The documentation on [Array Manipulation](https://docs.scipy.org/doc/numpy/reference/routines.array-manipulation.html) is a good one to keep bookmarked.
* `ndarray.reshape` creates a view with a new shape
* You can use `-1` as a value to infer the missing dimension
* `ndarray.ravel` returns a single dimensional view of the array.
* `ndarray.flatten` can be used to make a single dimensional copy.
* `np.lookfor` is great for searching docstrings from within a notebook.
```

[MUSIC] 0:00 You've done an excellent job learning all about NumPy arrays. 0:04 You can create, inspect, and modify them. 0:08 You can manipulate them into new shapes, and 0:10 you can filter out the values you don't want. 0:12 And now, I'd like to take some to show off some common use cases for 0:14 these arrays that we've been getting to know. 0:17 But before we dive in here, 0:20 I'd like to take a second to discuss a phenomenon known as imposter syndrome. 0:21 Imposter syndrome happens when you doubt your skills, and 0:28 you feel like you really aren't a pro. 0:30 You feel like a fraud. 0:32 You feel like everyone else knows more than you. 0:33 It's an unfortunate and quite common feeling, and 0:36 I want you to know that you aren't alone in experiencing it. 0:38 And I especially want you to know that you aren't actually a fraud and 0:41 that you can do this. 0:45 Because of the numerous used cases of NumPy arrays, you're bound to 0:46 see lots of math equations and formulas that you don't yet understand. 0:49 You might have learned once and then forgot. 0:54 What happens is you think, man I only know this much stuff, 0:56 look how much everyone else knows. 0:59 But here's the thing, very few people know it all. 1:01 They all know their own little bit, and you can and 1:04 will lean the extra little bits when you actually need them. 1:08 This is important to remember. 1:10 That feeling is bound to happen to you and I just want you to be ready for it. 1:12 Approach it like, hey, I'm not sure how to do that, yet, cuz you will learn it. 1:16 And look at you, you're doing great already. 1:20 I want you to approach things with the growth mindset. 1:23 Check the teacher's notes for more of that. 1:25 We are going to be doing a different type of programming than we've been 1:27 doing up until now on our Python journey together. 1:30 We are going to be doing what is known as Array Programming. 1:33 Instead of our typical way of looping through a collection of data and 1:36 pulling out values one by one to perform operations, 1:40 what we do in array programming is to perform the operation on 1:42 the entire set of values, all at once, in one fell swoop. 1:45 Not only is it faster, it's also expressed much more concisely in code. 1:49 Array programming code often ends up looking a lot like a math equation. 1:54 We've talked about how an array is referred to as a vector and 1:57 a two dimensional array as a matrix. 2:01 Well the reason for 2:03 that is because we use these arrays to represent their mathematical equivalent. 2:04 It's probably a little too abstract. 2:09 Let me show you what I mean by that statement. 2:11 Well, I've got an idea, let's do some linear algebra. 2:13 Why the long face? 2:17 That doesn't sound fun? 2:18 Did you forget how to do it too? 2:20 Don't worry. 2:21 All we have to do is set up the equation properly using the correct layout, and 2:22 NumPy is going to do all of the work for us. 2:27 I had a moment recently where I had to take back one of those famous quips that 2:29 we all make at one point in our lives, regarding stuff we're learning. 2:32 You know the one. 2:36 When am I ever going to need this? 2:36 Well, probably not very surprising to you by now, but this example involves tacos. 2:39 There's a new taco food cart that just opened by our office, and 2:43 I like to share my love of tacos with my coworkers. 2:46 So I often offer to do pickups for the office. 2:49 And if you've ever been in charge of a taco run, you know that one of the main 2:52 problems is that you're never quite sure of the price of the items. 2:55 So I'm usually just like, just pay me later, I'll put on my credit card. 2:58 And sometimes determining that price from the total is easy. 3:02 Like in the case where you have two tacos and the total is $3. 3:05 You can do that kind of math in your head, right? 3:08 So you might not even know how you did it, but 3:11 you know that a single taco costs $1.50. 3:13 That's pretty simple if that was the only item in the order, 3:16 but that's never the case, right? 3:19 So I know one of the larger orders I bought. 3:21 It was four tacos, one burrito, two horchatas and two Mexican cokes, 3:23 and the total was $20.50. 3:26 Nothing is better than a Mexi-Coke in a bottle, real sugar. 3:28 With that information though, 3:34 I have no idea how to figure out the price breakdown. 3:35 It's definitely not as that two tacos equation that we did in our head. 3:38 And this is because there are so many other variables now. 3:41 But I seem to remember from math, that if you have enough equations, 3:44 you should be able to solve them. 3:48 So let's see, do I have more data? 3:50 [LAUGH] Of course I do. 3:51 I went once, and I got a burrito and a Mexi-Coke, and that was 10 bucks. 3:53 And another office pick, if I get six tacos, one horchata, 3:57 and two Mexi-Cokes, and that was $14.25. 4:00 Now, one thing I remember from math is we're going to need to make sure that 4:03 everything is lined up. 4:07 That means I need to make sure that I enter a 0 for 4:08 all the items I didn't purchase, and we get some nice looking equations here. 4:11 Now, if that doesn't look like a math equation to you, 4:15 why don't we change those variables to their typical looking form. 4:18 Don't let those freak you out though. 4:21 Remember, they're just representing tacos and burritos and whatnot. 4:23 So, really this left side here kind of looks like rows and 4:26 columns almost, doesn't it? 4:28 In fact, let's make the type of food the heading here. 4:30 And what we have left here is the number of items, or the coefficient. 4:33 In mathematics, to denote a matrix, you will see hard brackets like this. 4:37 And for our totals, we can consider this as a single dimensional ray, 4:41 or as it's called, a vector. 4:45 A row base vector to be precise. 4:47 And with our systems of equations as a matrix, and our totals as a vector, 4:50 we should be able to solve for it. 4:53 There are definitely rules and 4:56 ways to go about solving that system of linear equations. 4:57 But what do you say we let NumPy do the work for us? 5:00 In the teacher notes, attached to this video, 5:03 I've added the equations to create our matrix. 5:05 So I'm gonna copy and paste it here in this cell. 5:07 So here is our orders array, and that's our matrix, right, 5:10 that's what it looks like. 5:13 It's an array with a rank of two, it's got two dimensions, and 5:14 each row here represents an order, right? 5:17 And this is the number of tacos, this first column here is tacos, right. 5:21 And then the next one is, next one being burritos, and 5:27 then horchata, and coke, right. 5:32 And now we need to get our constant prices. 5:35 So that's also in the teacher's notes. 5:38 Let me just paste that here. 5:39 So these are the totals of what came across. 5:40 Scroll this up a little bit. 5:47 Here we go, so we get our $3, 20.50 for the second order, 10, and 14.25. 5:49 So really what we've done here is to build the math equation, right? 5:54 Well, now for the awesome part. 6:00 We don't actually need to know how to solve this, we just need to know that it's 6:01 a system of linear equations that we're trying to solve for. 6:05 And we also need to know that there is a module in 6:08 the NumPy library for linear algebra. 6:12 It's shortened to linalg. 6:16 And here is the beautiful method. 6:17 Solve, and we pass [LAUGH] in our coefficient matrix, 6:22 which was our orders, and we also pass it our totals. 6:24 So we say totals like so. 6:29 And then let's go ahead and take a look at what prices is. 6:31 [LAUGH] Look at that, $1.5 tacos, we knew that, 6:34 8 bucks burritos, $1.25 horchatas, and a Mexican Coke for 2 bucks. 6:39 [LAUGH] Please note here that coke is more expensive than a taco already. 6:44 But well worth it. 6:49 Wait second, if we have this solution, 6:50 I think we should be able to verify it, right. 6:53 I mean, if we just plugged in these prices to the equations, 6:55 it should produce a total, right? 6:59 Shouldn't it? 7:02 So let's see. 7:03 Naturally in Python, I guess we would just loop through the matrix and then loop over 7:04 the prices and multiply them together, and then we'd add them together, right? 7:09 And that should produce the price. 7:15 Now, we could definitely do that, but let's remember that 7:17 one of the benefits of array programming is that we don't need to rely on loops. 7:21 Also, if there's a mathematical way to do something, 7:25 most likely there's a way to do it in NumPy. 7:27 Now what we are talking about here, multiplying matrices with vectors, and 7:29 then summing them together is actually a common math practice. 7:33 It is referred to as the inner product, and it's usually represented with a dot. 7:36 So you would say, A.B, that's exactly what we would've done in a loop but 7:40 expressed in mathematical terms, the notation, right? 7:46 So in Python though, that dot that I just created, that doesn't have any meaning. 7:50 So instead, we can use the at sign, like so. 7:53 So if we say orders @ prices, 7:58 let's see what happens. 8:03 So check that out. 8:07 That is what our totals are. 8:08 Look at that, it adds up. 8:09 So it took these and 8:10 ran them through the equations and ended up with the same totals. 8:11 So we know that it works. 8:15 And so really this is shorthand for 8:17 orders.dot (prices), see it's the same thing. 8:20 Pretty cool how we can use these shortcuts to build and solve equations, right? 8:25 It's super powerful, and 8:30 I'm glad that you have all those array manipulation powers in your bag of tricks. 8:31 As I know you're aware, 8:36 linear algebra has many more applications than just taco price calculation. 8:37 It's great for times, like we just saw, when you're trying to solve for 8:41 missing variables in a bunch of equations. 8:45 And remember, this is just one formula. 8:47 There are so many more. 8:50 And many of those formulas that are available for 8:52 you to use will most likely not be familiar. 8:54 Remember, that's okay. 8:57 No one knows them all. 8:59 You'll learn them as you need them. 9:00 Don't let the options overwhelm you. 9:02 Let them make you feel empowered. 9:03 You don't need to know how to solve them yourself, and that should feel great. 9:05 That linalg.solve function is a bit of an abstraction. 9:10 It really is doing all that work that you would have had to do by hand, 9:13 like you might have done in math class. 9:16 It's very similar to how we could've written that dot loop. 9:18 There's actually a function that does all the heavy lifting for us. 9:22 What do you say we take a quick look at some more common vector based operations 9:25 that you'll encounter as you progress. 9:29 Let me show you off NumPy's universal functions or ufuncs. 9:31 First though, let's take a quick break. 9:36 And why don't you jot down some thoughts and notes about what we just accomplished, 9:37 all without using the loops. 9:41 Yeah, quick reminder, make sure to check out the teacher's notes on this video. 9:42 I've dropped a lot of information about where to learn more about linear algebra, 9:46 if that got you re-invigorated. 9:49 And like I said, it's no big deal if it didn't. 9:51 There are many more use cases for NumPy. 9:54 See you soon and we'll compare notes. 9:57

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