Part 02 (Transcript)

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Er ... Geometry gives meaning to these concepts.

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So what we can think about geometrically ...

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... is a function from a plane to a plane ...

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... as a way to transform a plane.

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So we can think of transformation ...

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... that takes a point of the plane to another point of the plane.

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So what are the examples?

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Examples of transformations.

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Well let's take a piece of paper. Imagine that ...

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... it's not just a piece of paper, it's a whole plane.

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And what is it that we can do with that plane?

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Well the simplest thing we can do is we can shift it.

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Right? So we shift it for example to the right.

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And we imagine the whole plane being shifted.

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And each point is going to the right by the same amount.

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So that's ...

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... the first example.

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Let's call it translation.

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Geometrically translation is a motion of a plane that ...

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... moves every point in the same direction by the same amount.

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Now what else? Well you can think about rotation.

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Fix a point. And rotate by a certain angle.

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Right? So what happens is that one point in the center stays the same.

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And any other point is going to rotate about the center by a certain degree.

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By certain angle. And that angle of course is the same for all the points.

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So some points move just by a little bit.

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And some other points move a lot. And the further you go from the center ...

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... the more you have of that motion.

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So that's another example.

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And well, what else can you do?

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Well you can also ...

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Let's give this more exotic example.

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You can also for example take a line in the plane.

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This line.

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And reflect.

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Reflect the bottom pert of the plane to the top.

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So that the upper part of the plane stays the same. All the points here don't move anywhere.

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But all the points from below ...

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... flip and find their new home in the upper part of the plane.

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So this is the ... another example of transformation. Bum, reflection.

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Now, what else?

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[...] more complicated examples.

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Well, you can take the plane -- the whole plane.

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And can do something to it.

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And then ...

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... you'll smash it.

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Make it flat, totally flat.

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And then each point of the plane ...

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... again finds its new home in that plane.

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So somehow any point of the plane gets ...

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... transformed into another point of the plane by some rule. Right? So ...

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The reasonable geometric rule of how you translate the points ...

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... transform, well. Far from being translation, or rotation, or reflection.

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But well, there is a rule.

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Now, how to ...

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... relate this geometry to algebra?

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Well you can introduce coordinate system in the plane.

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And you can think about a point having coordinates x, y.

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And then applying ...

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... the transformation to that point results in a point ...

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... with coordinates u, v ...

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... of the same plane. Right?

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So it's ... er ...

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It's about taking a point to a point.

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Now er ... it's important to remind here about this concept of transformation and about concept of a function.

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Is that you should not think of this function as a process ...

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... because er ... a lot of examples here ...

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... many examples seem to be a process like translation or rotation.

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Well function is not a process. Er ...

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Let's recall the concept of a function ...

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... from Calc I when you have one variable x, one variable y and ...

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... y equals f of x is a function.

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Well you never think about this function as ...

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... a process of transforming this line ...

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... bending it and then smashing to the y-line.

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You think of the function as a process of taking a point here -- x ...

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... and then following some rule to result in a point there -- y.

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Right? You go straight -- here, there.

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Given x, result y.

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There is no process of transforming that you usually think about.

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And I'd like you not to think about any process.

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So translation, yes, it can be thought of as a process of moving point slowly to the right.

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But we will be interested in just going from the beginning straight to the end ...

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... without thinking about the process of motion.

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Or without thinking about the process of rotating.

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Original point -- end point. That's it. No process.

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Er ...

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