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