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Give me 30 minutes.
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Just 30 minutes.
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And I will make linear algebra click.
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Forever.
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You've probably seen these scary words before.
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Dot product, Gaussian elimination, vector space, determinants.
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They look like a foreign language.
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Maybe from some textbook you were forced to read and immediately forgot.
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But this isn't going to be some dry lecture.
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We're going to build your intuition from the ground up.
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And the best part?
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we're going to show you how every single one of these concepts connects.
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Okay, take a look at this table on the screen.
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This is our entire journey.
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It's our roadmap.
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Take a look at this table.
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We'll start with vectors and the dot product, measuring similarity.
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Then linear systems, ax equals b, solving complex problems.
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Vector spaces, span, basis, the rules of your world.
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And linear transformations, where matrices become actions.
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plus determinants.
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One number that reveals everything.
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I get it.
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Right now, looking at all of this, it might look intimidating.
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But here is my promise to you.
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By the end of this video, every single row you see on that screen will make perfect intuitive sense.
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Now it's time to tackle that first row.
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We're starting with the most fundamental object in this entire field.
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The vector.
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But forget arrows in a physics textbook.
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That's not what we're talking about today.
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Let's ask a real question.
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How does a streaming service know you'll like a movie?
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How can it possibly quantify something as fuzzy as personal taste?
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It all starts with representing your taste as a vector.
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Let's look at the movie ratings example.
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Imagine three users and their ratings for three movies.
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We've got sci -fi, comedy, and romance.
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User A gives us the numbers 5, 4, 1.
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They love sci -fi, they like comedy, but they hate romance.
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User B gives us 4, 5, 2.
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Pretty similar to user A.
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And user C gives us 1, 2, 5.
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Very different.
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Right now, these are just lists of numbers.
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But if we treat them as coordinates in a 3D movie space, something magical happens.
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Forget the math for a second.
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Seriously, just look at this picture.
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You can see the user taste in this 3D movie space.
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Here is the key idea.
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The smaller the angle between two vectors, the more similar they are.
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That's it.
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That's the core intuition.
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Our entire problem now is just to calculate that angle.
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The tool we use is called the dot product.
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Now, the dot product has two formulas but one goal.
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Here's the secret.
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We use the easy one, the calculation one, to solve for the hard one, the geometry one.
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First, the calculation formula.
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This is easy.
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You just multiply corresponding components and add them up.
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a .b equals a1b1 plus a2b2 plus a3b3.
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Simple.
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Second, the geometric formula.
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This connects the dot product to the angle, theta.
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a .b equals the length of a times the length of b times the cosine of theta.
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And that length, by the way, is just the square root of a1 squared plus a2 squared plus a3 squared.
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So here's the game plan.
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1. We calculate A .B using the easy calculation formula.
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2. We calculate the lengths of A and B. And 3.
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We plug those numbers into the geometric formula to solve for the angle theta.
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Let's do it.
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Our user vectors are A equals 5, 4, 1, B equals 4, 5, 2, and C equals 1, 2, 5.
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Case 1.
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Comparing similar users A and B.
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Step 1.
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Calculate the dot product A .B.
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We take 5 times 4 plus 4 times 5 plus 1 times 2, which equals 42.
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Okay, step 2.
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Calculate the vector lengths.
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The length of A is the square root of 5 squared plus 4 squared plus 1 squared.
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That's the square root of 25 plus 16 plus 1, which is the square root of 42.
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About 6 .48.
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The length of B is the square root of 4 squared plus 5 squared plus 2 squared.
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That's 16 plus 25 plus 4, square root of 45.
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About 6 .71.
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Almost there.
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Step 3.
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Solve for the angle theta.
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We rearrange the formula.
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Cosine theta equals a dot b divided by the length of a times the length of b.
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So that's 42 divided by 6 .48 times 6 .71, which gives us about 0 .966.
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So theta is the arc cosine of 0 .966, which is approximately 15 degrees.
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That's tiny.
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The math just proved what our eyes told us.
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These users have very similar tastes.
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Okay, case two.
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Comparing dissimilar users A and C.
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Quick quiz.
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What do you expect?
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A bigger angle or a smaller one?
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Let's find out.
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Our user vectors are a, which is 5, 4, 1, and c, which is 1, 2, 5.
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Step 1, the dot product, a .c, equals 18.
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Step 2, the length of c, that's the square root of 1 squared plus 2 squared plus 5 squared, which is about 5 .48.
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Step 3.
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Solve for the angle.
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Cosine theta equals 18 divided by 6 .48 times 5 .48.
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That comes out to about 0 .507.
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So theta is the arc cosine of that, which is 59 .5 degrees.
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A much wider angle.
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The number 59 .5 degrees is our precise mathematical measure of how different their tastes are.
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You just understood the entire theory.
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Now, let's get some instant gratification with code.
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We import numpy, we define our user ratings.
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A is a numpy array of 5, 4, 1.
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B is 4, 5, 2.
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And C is 1, 2, 5.
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Now the magic is in these two lines.
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Watch this.
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The dot product for A and B is just np .dot with A and B.
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Then the cosine of theta is
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that dot product divided by the norm of A times the norm of B And then we just convert that to degrees
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We do the same thing for A and C And when we print it out
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The angle between A and B is 15 .0 degrees
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The angle between A and C is 59 .5 degrees
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The exact same results
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You've seen the intuition
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You've done the math
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You've seen the code
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The dot product is a machine for measuring similarity.
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So, we've handled single vectors.
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But what happens when we need to solve problems with systems of relationships all at once?
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This is where linear algebra was born.
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It wasn't born from abstract math, but from concrete problems.
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Let's look at one.
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The problem.
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Coffee blends.
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A coffee shop has two special blends and a limited stock of beans.
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Okay, so blend A requires 300 grams of Arabica and 100 grams of Robusta.
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Got it.
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Blend B is a bit different.
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It needs 100 grams of Arabica and 200 grams of Robusta.
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You look at your total stock, and you have 11 ,000 grams of Arabica and 8 ,000 grams of Robusta.
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So here's the question.
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How many units of each blend can you make to use up your entire stock?
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This is a classic linear algebra problem.
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Now, let's think about the high school way of doing this.
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If we make x units of blend A and y units of blend B, we'd write it out.
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For Arabica, it's 300x plus 100y equals 11 ,000.
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And for Robusta, it's 100x plus 200y equals 8 ,000.
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This is messy.
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Writing out equations like this is slow, and, frankly, it doesn't scale well.
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Linear algebra gives us a clean, powerful way to organize all of this.
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The blueprint AX equals B.
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Now, don't panic.
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This isn't a new rule you have to memorize.
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It's just a filing system, a way to keep things tidy.
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Let's look at the pieces.
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First, we have A.
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This is the recipes matrix.
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See here?
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The first column is the recipe for blend A, 300 and 100.
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The second column is for blend B, 100 and 200.
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Next is X.
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That's the what you want vector.
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It holds our unknowns, x and y.
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And finally, b.
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That's our stock vector.
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The total stuff we have, 11 ,000 and 8 ,000.
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Putting it all together, our entire problem is now just one clean line.
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The matrix with 300, 100, 100, 200 times the vector xy equals the vector 11 ,000, 8 ,000.
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This is the universal language for systems of linear equations.
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So how do we solve it?
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The algorithm is a game!
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It's called Gaussian elimination!
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Here's the game plan.
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The goal.
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Turn our matrix A into an upper triangle shape.
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This is called row echelon form.
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And the legal moves?
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There are only three.
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One, you can swap any two rows.
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Two, you can multiply an entire row by a non -zero number.
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And three, you can add a multiple of one row to another row.
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That's it.
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Let's play.
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Step one.
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Set up the game board.
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We combine A and B into what's called an augmented matrix.
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It looks like this.
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A 2x3 grid with 300, 100, and 11 ,000 on the top row and 100, 200, and 8 ,000 on the bottom.
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Step 2.
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Simplify the board.
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We can use rule 2.
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These numbers are huge.
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Let's divide the first row, R1, by 100, and let's divide the second row, R2, by 102.
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Look at that!
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We go from that big messy board to a new one.
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3 -1, 110 on top, and 1 -2, 80 on the bottom.
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Much cleaner.
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Step 3.
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Get a 1 in the top left pivot.
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We'll use rule 1 for this.
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It's always easier to work with a 1.
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So, let's just swap row 1 and row 2.
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And now, our board looks like this.
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1 -2, 80 on top, and 3 -1, 110 on the bottom.
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Perfect.
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Our first pivot is locked in.
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Step 4.
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Eliminate below the pivot.
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This is the key move using rule 3.
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We want to turn that 3 in the bottom left into a 0.
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So we'll make a new row 2 by taking the old row 2 and subtracting 3 times row 1 from it.
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Let's do the math.
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For the first element, 3 minus 3 times 1 is 0.
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For the second, 1 minus 3 times 1 is negative 2.
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2 is negative 5.
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And for the last one, 110 minus 3 times 80, which is negative 130.
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And we're done.
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We've won the game.
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Look at that beautiful upper triangle shape.
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This is row echelon form.
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Step 5.
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The payoff.
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It's called back substitution.
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We just turn this back into equations.
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The bottom row tells us 0x minus 5y equals negative 130.
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That's easy.
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Y equals negative 130 divided by negative 5, which is 26.
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Now we take the top row.
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1x plus 2y equals 80.
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We know y is 26, so we plug it in.
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x plus 2 times 26 equals 80, which means x is 28.
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The result?
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The shop can make 28 units of blend A and 26 units of blend B.
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You just solved a real -world system from scratch.
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You just did the hard work to build the intuition, and what's your reward?
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In the real world, you'll never do that by hand again.
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Your reward is the one -liner.
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We import numpy as np, we define our recipe matrix A as a numpy array with our values,
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we define our stock vector B, and then the magic.
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x equals np .lin alg .solve with A and B inside.
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We print the results, and...
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units of blend A, 28 .0, units of blend B, 26 .0.
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And now you know exactly what that one line of code is doing under the hood.
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So, we've solved a specific problem with a specific amount of beans.
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But what are the rules of this coffee -making world?
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What are all the possible combinations of blends we could have made?
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To answer that, we need to define the rules of the world our vectors live in.
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Now, I'll be honest, this chapter is the most abstract one we're going to cover.
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But I'm going to make it click with a simple analogy.
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Forget coffee.
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You're programming a 2D video game character.
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You give your character two move commands, move A and move B.
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Here are the two critical questions.
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First, can your character reach every single point on the screen with just those two moves?
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And second, are both moves actually necessary, or is one of them redundant?
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Answering these two questions is the entire point of this chapter.
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Let's build the core vocabulary of your world.
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The standard 2D grid uses two building block vectors to get anywhere.
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The first is I, which is 1, 0.
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That's just one step to the right.
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The second is J, which is 0, 1, or one step up.
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So to get to the point 3, 4, you just take a scaled sum.
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You take 3 times the right vector, plus 4 times the up vector, and you land on 3, 4.
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This is called a linear combination.
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Our game characters' moves are just different building blocks.
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So let's define our terms.
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First, span.
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This answers the question, where can my character go?
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The span is the set of all possible points you can reach.
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Does it cover just a tiny line, or does it cover the entire 2D plane?
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Second term, linear independence.
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This answers the question, are any of my moves redundant?
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If move B is just two times move A, they are linearly dependent.
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The second move is useless.
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It doesn't unlock any new directions.
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You're just stuck on the same path.
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Which brings us to our third term, basis.
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This is the perfect set of moves.
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A basis has to be two things.
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First, it's linearly independent, so there are no redundant moves.
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And second, it spans the entire space, so you can reach everywhere.
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A basis is the minimal, most efficient set of building blocks for your world.
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For our 2D game screen, any basis must have exactly two vectors.
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We say the dimension of the space is 2.
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Let's look at some movesets.
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You'll see the difference instantly.
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Okay, in this table we're going to look at four different sets of moves.
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We'll ask if they span the plane, if they're independent, and if they make a basis.
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Case 1, the standard basis.
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The vectors are 1, 0, and 0, 1.
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Does it span the 2D plane?
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Yes.
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Can it get everywhere?
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Is it linearly independent?
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Yes.
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No redundancy.
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So, is it a basis?
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Yes.
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It's perfectly efficient.
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Case 2.
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The set with vectors 1, 1, and 2, 2.
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Does it span the plane?
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No. You're totally stuck on a line.
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Are they independent?
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The second move is just twice the first.
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It's totally redundant.
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It adds nothing new.
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So it's definitely not a basis.
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What if we have three moves?
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1, 0, 0, 1, and 1, 1.
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Does it span the plane?
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Yes, you can certainly get everywhere.
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But is it independent?
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The third vector is just the first two added together.
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It's unnecessary.
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So it's not a basis because it's not efficient.
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It's bloated.
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Case 4, a non -standard basis.
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The vectors are 1, 2, and 3, 1.
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Does it span the plane?
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Yes, they point in different enough directions.
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Are they independent?
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Yes, you can't make one from the other.
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So is it a basis?
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Yes, it's a perfect, efficient, and complete set of moves, even though they're a bit skewed.
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Okay, you're thinking, I get the game analogy, But why should I care about a non -standard basis?
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Because changing your basis is a superpower.
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It lets you describe the same data from a more effective point of view.
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It's the core idea behind JPEG image compression.
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It's also the core idea behind
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Principal Component Analysis, or PCA.
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Let's connect this back to what you already know.
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Here's the problem.
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We have a point in the standard basis, P equals 7, 5.
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And we have our new skewed basis, b, made of the vectors 1, 2, and 3, 1.
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The question is, what are p's coordinates in our new basis?
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We're looking for two numbers.
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Let's call them c1 and c2, such that c1 times our first basis vector plus c2 times our second basis vector equals our point p.
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Does this look familiar?
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It's the exact same ax equals b problem from the last chapter.
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We can write it as a matrix equation.
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A matrix with our basis vectors as columns times a vector with our unknown coordinates c1
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and c2 equals our point vector p.
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We're just solving bc equals p.
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The concepts are connecting.
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Let's solve it.
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We import numpy as in p.
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Our new basis vectors form the columns of a matrix we'll call b.
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So b equals in p dot array with rows 1, 3, and 2, 1.
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The point p we want to express is an array with 7, 5.
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Then we solve the system BC equals P for C, which are the new coordinates.
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So C equals inP dot lin alg dot solve, passing in B and P.
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Let's print the coordinates of P in our new basis B.
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The output is 1 .6 and 1 .8.
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This means we've described the exact same point using a different set of building blocks.
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Our point P is just 1 .6 times the vector 1, 2, plus 1 .8 times the vector 3, 1.
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This isn't just a party trick, it's a foundational technique in modern data analysis and computer graphics.
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We just established the quote -unquote rules of the world our vectors live in.
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We talked about basis vectors and how they define our grid.
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But that world was static.
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It just sat there.
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Now, we're going to make things happen.
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And here is the single biggest mindset shift in all of linear algebra.
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Up until now, you've probably seen matrices as boring grids of numbers, something you use for solving A times X equals B.
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Forget that.
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Erase it from your mind.
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From this moment on, a matrix is an action.
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It's a verb.
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It rotates, scales, and shears space itself.
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The equation V' equals MV doesn't mean solve for X
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it means the matrix M acts on the vector V to produce a new transformed vector, V'.
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Let's make this real with a problem, rotating a game character.
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Imagine you've got this triangle on the screen.
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Its points before the rotation are P1 equals 11, P2 is 31, and P3 is 22.
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Your job is to rotate this triangle 90 degrees counterclockwise around the origin.
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How do you find the new coordinates?
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The after coordinates you want are P1' at minus 1, 1, P2' at minus 1, 3, and P3' at minus 2, 2.
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But how do you get there systematically?
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Here's the secret.
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The key to understanding any linear transformation.
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You only need to watch where the basis vectors i and j land.
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That's it.
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That's the whole trick.
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Let's watch it happen for our 90 degree rotation.
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Before the transformation, our basis vectors are the usual suspects.
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i is the vector pointing right, and j is the vector pointing straight up.
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Now, we apply the 90 degree rotation.
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What happens?
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i, which was at gets rotated and now it lands at .
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And J, which was at gets rotated and lands over at . And here's the magic.
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The columns of a transformation matrix are just the coordinates of where the original basis vectors land.
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So let's build our matrix.
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1. The first column is where I landed.
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That's the vector .
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2. The second column is where J landed.
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That's the vector . And BOOM!
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Looking at this matrix R, we have a first column of 0, 1, and a second column of minus 1, 0.
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This is the genetic code for a 90 degree rotation.
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Now for the payoff, let's rotate that triangle.
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Here's the setup.
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We have our rotation matrix R, and we have our three vertices, P1, P2, and P3.
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Step 1.
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Let's transform vertex P1.
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We calculate P1' equals R times P1, so that's our rotation matrix, times the vector 1 ,1.
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The result is the vector minus 1 ,1.
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So our new point P1' is at minus 1 ,1.
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Exactly what we wanted.
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Step 2.
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Transform vertex P2.
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Same deal.
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P2' is R times P2, our matrix, times the vector 3 ,1.
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And that gives us the vector minus 1 ,3.
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The new point P2' is at .
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Step 3.
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And finally, transform vertex P3.
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P3' is R times P3.
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That's our matrix times the vector .
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The result is .
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The new point P3' is at .
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Look at that!
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We have the before triangle and the after triangle.
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A perfect rotation!
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Now, from one triangle to millions.
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This isn't some abstract concept.
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It is the engine of every video game you have ever played.
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Your graphics card, your GPU, is a piece of hardware optimized for one primary job.
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Doing billions of v' equals mv multiplications per second.
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So let's see how the professionals do it. With code.
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First, the input.
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We pack all our vertices into a single matrix.
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We import numpy as np.
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Here's our 90 -degree rotation matrix, R, which is a numpy array with rows 0, minus 1, and 1, 0.
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Now here's the cool part.
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Our triangle vertices become the columns of a single matrix, P.
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Then, the calculation.
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It's just one operation to transform all the points.
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We write P underscore transformed equals R at P.
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That at symbol is Python's way of saying matrix multiplication.
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And the output?
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Perfect results.
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Instantly.
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We can print the original vertices and the transformed vertices.
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And look, the original matrix P has columns 1 1, 3 .1, and 2 .2.
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The transformed matrix has columns minus 1 1, minus 1 3, and minus 2 2.
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It's the exact same result we got by doing it one by one, but it's way more efficient.
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This is the power of thinking of matrices as actions.
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We saw that matrices are not just grids of numbers.
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They are verbs.
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They are actions that rotate, scale, and shear space.
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We saw our rotation matrix R spin a triangle without changing its size at all.
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But a scaling matrix?
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That would have blown it up.
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This leads to a critical question.
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Isn't there one single number that just tells you how much a matrix scales area?
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A single diagnostic number?
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There is.
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It's called the determinant.
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Okay, forget the scary formulas for a moment.
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Just for a moment.
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Here is the single most important idea of this entire chapter.
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The determinant is the area scaling factor of a transformation.
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The determinant of a 2x2 matrix is the area of the parallelogram that the 1x1 unit square gets transformed into.
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That's it.
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That's the whole story.
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As you can see in this little diagram, you start with a simple unit square with an area of 1.
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You apply a transformation, M, and it becomes a parallelogram.
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The area of that new parallelogram is the absolute value of the determinant of M.
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This one number tells you everything.
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For example, if the determinant of m is 1, then areas are perfectly preserved.
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This is what you see in a pure rotation.
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If the determinant of m is 2, the transformation doubles all areas.
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And what if the determinant of m is less than 0, say, negative 1?
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This means the orientation of space gets flipped.
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It's like looking in a mirror.
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And now, for the most important case of all.
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If the determinant of m is zero, the area is zero.
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The matrix has squished your entire world into a line, or maybe even just a single point.
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You've lost a dimension, and the transformation is irreversible.
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You can't go back.
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This matrix is called singular, or non -invertible.
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Okay, so let's see the algorithm.
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For a 2x2 matrix, the calculation is actually super easy.
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For a matrix A with elements A, B, C, D, the formula is right here.
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The determinant of A equals A, D, minus B, C.
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Alright, time for some rapid -fire diagnostics.
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Let's analyze three transformations using just this one number.
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First up, case 1, the 90 -degree rotation matrix.
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The matrix R has rows 0, minus 1, and 1, 0.
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Let's do the calculation.
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The determinant of R is 0 times 0 minus minus 1 times 1.
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So that's 0 minus negative 1, 1.
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The result?
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The determinant of R is 1.
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The math confirms it.
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Pure rotation preserves area perfectly.
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You can see it here.
501
The unit square just spins, but its area stays exactly 1.
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Next, case 2, a scaling matrix.
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Our matrix S has rows 2, 0, and 0, 3.
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The calculation is 2 times 3 minus 0 times 0.
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That's 6 minus 0.
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The result?
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The determinant of S is 6.
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This means this transformation multiplies all areas by exactly 6.
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The unit square, which started with an area of 1, becomes a rectangle with an area of 6.
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Area multiplied by 6.
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And finally, case 3, the broken singular matrix.
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Look at this matrix C.
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The determinant of C is 0.
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This is our red flag.
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This matrix collapses 2D space onto a line.
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As you see here, the unit square gets completely squashed.
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It has an area of 0.
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Space collapsed.
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So here's your diagnostic toolkit.
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These properties are incredibly useful.
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If the determinant of A is not 0, then A is invertible.
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The transformation can be reversed.
523
If the determinant of A is 0, then A is singular.
524
Space is squashed into a lower dimension.
525
Here's a cool one.
526
The determinant of A times B is just the determinant of A times the determinant of B.
527
So the scaling factors just multiply together.
528
And that means the determinant of A inverse is just 1 over the determinant of A.
529
It just reverses the scaling.
530
Remember our coffee problem from earlier?
531
ax equals b?
532
Before you even try to solve it, you can calculate the determinant of a.
533
If it's zero, you know immediately the system is broken.
534
Of course, in the real world, you can do this with a one -second check with code.
535
Let's see it.
536
First, the input.
537
We import numpy as np.
538
We create our three arrays.
539
R for rotation, S for scaling, and C for the collapsing matrix.
540
Then, the calculation.
541
It's super simple.
542
We create a variable det underscore r and set it to np .linalg .det of r, and we do the same for s and c.
543
And the output?
544
The determinant of r is 1, the determinant of s is 6, and the determinant of c is 0.
545
The results match perfectly.
546
The determinant is your matrix's vital sign, giving you an instant health check on your transformations.
547
And just like that, you did it.
548
Let's look at the journey we took.
549
The analogy became the concept.
550
You went from movie ratings to the dot product, understanding it as similarity.
551
From delicious coffee blends to solving systems of equations with AX equals B.
552
You went from video game moves to understanding span and basis as the fundamental rules of the world.
553
And finally, you saw that matrices are verbs, the powerful engine behind all computer graphics.
554
But here's the most important part.
555
This isn't the end.
556
It's the foundation for everything that comes next.
557
Things like eigenvectors, PCA, SVD, the master algorithms of machine learning and data science.
558
And because of the work you just put in, you are now ready to understand them.
559
Truly.
560
Linear Algebra has finally clicked for you.
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Vocabolario e note di pronuncia per questa lezione
Questa lezione di conversazione si basa sul video “Give Me 30 min, I will make Linear Algebra Click Forever”. Le parole che ritornano più spesso: vector, matrix, equals, determinant, basis. Questo video contiene 560 frasi e 4294 parole da ripetere con lo shadowing. Il parlato dura 29:56. Chi parla mantiene un ritmo costante di circa 143 parole al minuto, comodo per lo shadowing. Solo il 74% delle parole rientra nelle 3.000 più comuni dell’inglese, quindi il lessico è impegnativo.
Vocaboli chiave di questo video
15 parole del video che vale la pena imparare, con pronuncia e significato:
| Parola | Pronuncia | Significato |
|---|---|---|
| vector sostantivo | /ˈvɛktɚ/ | vettore |
| minus | /ˈmaɪ.nəs/ | meno |
| solve verbo | /sɒlv/ | risolvere |
| blend verbo | /blɛnd/ | mescolare, amalgamare |
| rotation sostantivo | /ɹoʊˈteɪʃən/ | rotazione, giro completo |
| linear aggettivo | /ˈlɪn.i.əɹ/ | lineare |
| spin verbo | /spɪn/ | girare, far girare |
| angle sostantivo | /ˈæŋ.ɡəl/ | angolo |
| triangle sostantivo | /ˈtɹaɪˌæŋɡəl/ | triangolo |
| transform verbo | /tɹænsˈfɔɹm/ | trasformare, modificare |
| formula sostantivo | /ˈfɔː.mjʊ.lə/ | formula |
| coordinate verbo | /koʊˈɔɹ.dəˌneɪt/ | coordinare, accordarsi |
| calculation sostantivo | /ˌkælkjuˈleɪʃn̩/ | conto, calcolo |
| divide verbo | /dɪˈvaɪd/ | dividere, dividersi |
| calculate verbo | /ˈkælkjʊleɪt/ | calcolare |
I phrasal verb che sentirai
| Parola | Pronuncia | Significato |
|---|---|---|
| call back verbo | richiamare | |
| find out verbo | scoprire | |
| go back verbo | tornare | |
| set up verbo | /ˌsɛt ˈʌp/ | preparare, installare |
| write out verbo | appuntare, emettere |
Frasi da ripetere
Frasi brevi e complete del video che puoi riutilizzare nella conversazione di tutti i giorni:
- How does a streaming service know you'll like a movie?
- This answers the question, where can my character go?
- What if we have three moves?
- The question is, what are p's coordinates in our new basis?
- How do you find the new coordinates?
La grammatica di questo video
Le strutture che chi parla usa di più, con le parole esatte del video:
| Struttura | Nel video |
|---|---|
| Present perfect have/has + participio passato — un’azione passata che conta ancora adesso | You've probably seen · You've seen · You've done |
| Forma passiva be + participio passato — conta ciò che accade, non chi lo fa | were forced · is called · was born |
| Frasi relative who / which + frase — un’informazione in più su una persona o una cosa | squared, which is · that, which is |
Pronuncia a cui fare attenzione
Chi parla usa 52 contrazioni e forme ridotte, come we're, you've, isn't. Pronunciale nella forma breve, così come le senti.
- I suoni “th”: mathematics /mæθ(.ə)ˈmæt.ɪks/, algorithm /ˈælɡəɹɪðm̩/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, theta /ˈθiːtə/
- I suoni “sh” e “zh”: rotation /ɹoʊˈteɪʃən/, transformation /ˌtɹæns.fɔɹˈmeɪ.ʃən/, calculation /ˌkælkjuˈleɪʃn̩/, equation /ɪˈkweɪ.ʒən/, efficient /ɪˈfɪʃənt/
- Parole lunghe — attenzione all’accento: transformation /ˌtɹæns.fɔɹˈmeɪ.ʃən/, coordinate /koʊˈɔɹ.dəˌneɪt/, mathematics /mæθ(.ə)ˈmæt.ɪks/, intuition /ˌɪn.tjuːˈɪʃ.ən/, analogy /əˈnæləd͡ʒi/
I suoni difficili per chi parla italiano:
- Consonante finale — senza aggiungere una vocale dopo: matrix /ˈmeɪ.tɹɪks/, minus /ˈmaɪ.nəs/, solve /sɒlv/, blend /blɛnd/, coordinate /koʊˈɔɹ.dəˌneɪt/
- /æ/ — più aperta della “e”: transformation /ˌtɹæns.fɔɹˈmeɪ.ʃən/, angle /ˈæŋ.ɡəl/, triangle /ˈtɹaɪˌæŋɡəl/, transform /tɹænsˈfɔɹm/, calculation /ˌkælkjuˈleɪʃn̩/
Come esercitarsi con questo video
- Ascolta tutto il video una volta senza parlare e annota le parole che non conosci.
- Fai shadowing frase per frase a velocità normale, ripetendo ognuna finché il tuo ritmo coincide con quello di chi parla.
- Registrati e confronta con l’originale, facendo attenzione a parole come vector, minus, solve.
Cos'è la tecnica dello Shadowing?
Shadowing è una tecnica di apprendimento delle lingue supportata da studi scientifici, originariamente sviluppata per la formazione dei traduttori professionisti e resa popolare dal poliglotta Dr. Alexander Arguelles. Il metodo è semplice ma potente: ascolti un audio in inglese di madrelingua e lo ripeti immediatamente ad alta voce — come un'ombra che segue il parlante con un ritardo di solo 1–2 secondi. A differenza dell'ascolto passivo o degli esercizi di grammatica, lo shadowing costringe il tuo cervello e i muscoli della bocca a elaborare e riprodurre simultaneamente i modelli di discorso reale. La ricerca dimostra che migliora significativamente la precisione della pronuncia, l'intonazione, il ritmo, il discorso connesso, la comprensione dell'ascolto e la fluidità del parlato — rendendolo uno dei metodi più efficaci per la preparazione alla prova di speaking dell'IELTS e per la comunicazione reale in inglese.