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Current time:0:00Total duration:5:16

so I have talked a lot about different ways that you can visualize multivariable functions functions that'll have some kind of multi-dimensional input or output these include three dimensional graphs which are very common contour maps vector fields parametric functions but here I want to talk about one of my all-time favorite ways to think about functions which is as a transformation so anytime you have some sort of function if you're thinking very abstractly I like to think that there's some sort of input space and I'll draw it as a blob even though you know that could be the real number line so it should be a line or it could be three dimensional space and then there's some kind of output space and again I just very vaguely think about it as this blob but that could be again the real number line the xy-plane three dimensional space and the function is just some way of taking inputs to outputs and every time that we're trying to visualize something like with a graph or a contour map you're just trying to associate input-output pairs you know if f inputs you know three gets mapped to the vector one two it's a question of how do you associate the number three with that vector one two and the thought behind transformations is that we're just going to watch the actual points of the input space move to the output space and I'll start with a simple example it's just a one-dimensional function it'll have a single variable input and it will have a single variable output so let's consider the function f of X is equal to x squared minus three and of course the way we're used to visualizing something like this it'll be as a graph you might kind of be thinking of something roughly parabolic that's squished down by three but here I don't want to think in terms of graphs I just want to say how do the inputs move to those outputs so as an example if you go to zero when you plug in zero you're going to get negative three you know zero squared minus three is equal to negative three so somehow we want to watch zero move to negative three and then similarly let's say you plug in one and you get one squared minus three is negative two so somehow we want to watch one move to negative two just to list another example here let's say you're plugging in three itself so three squared minus 3 is 9 minus 3 is 6 so somehow in this transformation we want to watch 3 move to the number 6 and with the little animation we can watch this happen we can actually watch what it looks like for all these numbers to move to their corresponding outputs so here we go each number will move over and land on its output and I'll clear up the board here so I kept track of what the original input numbers are by just kind of writing them on top here and that was a way of just watching how it moves and I'll play it again here let's let's just watch where each number from the input space moves over to the output and with single variable functions this is a little bit nice because it gives this sense of inputs moving to outputs but where it gets fun is in the context of multiply variable functions so now let me consider a function that has a one-dimensional input and a two-dimensional output and specifically it'll be f of X is equal to cosine of X cosine of X and then the Y component will be X sine of Y sorry X sine of X so just just to think about it examples if you plug in something like zero and think about where zero ought to go you would have F of zero is equal to cosine of 0 is 1 and then 0 times anything is zero so somehow we're going to watch zero move over to the point 1 0 and so this is where we expect zero to land and let's think about like pi so f of pi then cosine of PI is negative 1 this is going to be PI x and sine of pi is 0 so that will again be 0 so you know this little guy is where 0 lands and we expect that this is going to be where the value PI lands and if we watch this take place and we actually watch each element of the input space move over to the output space we get something like this and again this is just kind of a nice way to think about what's actually going on you might ask questions about whether the spacings of getting scratches stretched or squished and notice that this is also what a parametric plot of this function would look like if you interpret it as a parametric function this is what you get in the end but whereas in parametric spots you lose input information here you can kind of see where things move as you go from one to the other and in the next video I'm going to talk about how you can interpret functions with a two dimensional input and a two dimensional output as a transformation