Add colorbar to existing axis

I'm making some interactive plots and I would like to add a colorbar legend. I don't want the colorbar to be in its own axes, so I want to add it to the existing axes. I'm having difficulties doing this, as most of the example code I have found creates a new axes for the colorbar.

I have tried the following code using matplotlib.colorbar.ColorbarBase, which adds a colorbar to an existing axes, but it gives me strange results and I can't figure out how to specify attributes of the colorbar (for instance, where on the axes it is placed and what size it is)

import matplotlib
import matplotlib.pyplot as plt
from matplotlib.cm import coolwarm
import numpy as np

x = np.random.uniform(1, 10, 10)
y = np.random.uniform(1, 10, 10)
v = np.random.uniform(1, 10, 10)

fig, ax = plt.subplots()

s = ax.scatter(x, y, c=v, cmap=coolwarm)

matplotlib.colorbar.ColorbarBase(ax=ax, cmap=coolwarm, values=sorted(v),
                                 orientation="horizontal")

Using fig.colorbar instead ofmatplotlib.colorbar.ColorbarBase still doesn't give me quite what I want, and I still don't know how to adjust the attributes of the colorbar.

fig.colorbar(s, ax=ax, cax=ax)

Let's say I want to have the colorbar in the top left corner, stretching about halfway across the top of the plot. How would I go about doing that?

Am I better off writing a custom function for this, maybe using LineCollection?

The colorbar has to have its own axes. However, you can create an axes that overlaps with the previous one. Then use the cax kwarg to tell fig.colorbar to use the new axes.

For example:

import numpy as np
import matplotlib.pyplot as plt

data = np.arange(100, 0, -1).reshape(10, 10)

fig, ax = plt.subplots()
cax = fig.add_axes([0.27, 0.8, 0.5, 0.05])

im = ax.imshow(data, cmap='gist_earth')
fig.colorbar(im, cax=cax, orientation='horizontal')
plt.show()