Now actually, if we could find what the steepest way down the hill was,

then we could go down this set of contours, this sort of landscape here

towards the minimum point, towards the point where get the best possible fit.

And what we're doing here,

these are vectors, these are little moves around space.

They're not moves around a physical space,

they're moves around a parameter space, but it's the same thing.

So if we understand vectors and we understand how to get down hills,

that sort of curviness of this value of goodness, that's calculus.

Then once we got calculus and

vectors, we'll be able to solve this sort of problem.

So we can see that vectors don't have to be just geometric objects in the physical

order of space.

They can describe directions along any sorts of axes.

So we can think of vectors as just being lists.

If we thought of the space of all possible cars, for example.

So here's a car.

There's its back, there's its window, there's the front, something like that.

There's a car, there's the window.

We could write down in a vector all of the things about the car.

We could write down its cost in euros.

We could write down its emissions performance in grams of CO2 per

100 kilometers.

We could write down its Nox performance,

how much it polluted our city and killed people due to air pollution.

We could write down its Euro NCAP star rating, how good it was in a crash.

We could write down its top speed.

And write those all down in a list that was a vector.

That'd be more of a computer science view of vectors,

whereas the spatial view is more familiar from physics.

In my field, metallurgy, I could think of any alloy as being described by a vector

that describes all of the possible components,

all the compositions of that alloy.

Einstein, when he conceived relativity,

conceived of time as just being another dimension.

So space-time is a four dimensional space, three dimension of metres, and

one of time in seconds.

And he wrote those down as a vector of space-time of x, y, z,

and time which he called space-time.

When we put it like that, it's not so

crazy to think of the space of all the fitting parameters of a function, and

then of vectors as being things that take us around that space.

And what we're trying to do then is find the location in that space,

where the badness is minimized, the goodness is maximized, and

the function fits the data best.

If the badness surface here was like a contour map of a landscape, we're trying

to find the bottom of the hill, the lowest possible point in the landscape.

So to do this well, we'll want to understand how to work with vectors and

then how to do calculus on those vectors in order to find

gradients in these contour maps and minima and all those sorts of things.

Then we'll be able to go and do optimizations, enabling us to go and

work with data and do machine learning and data science.