attempt to explain why broadcasting is bae

## broadcast.Rmd
A starter example is centering a matrix

```{python}
import numpy as np

X = np.random.normal(size=(3, 3))  # random 3 by 3 matrix

mu = X.mean(axis = 0)    # array (vector) of column means

print("Original matrix")
print(X)

print("\nArray of column means")
print(mu)

# array effectively copied till vector matches matrix in size
print("\nBroadcast subtraction operation")
print(X - mu)
```

To do this in R you have to loop or use something like map/apply.

This lets you create kernel matrices in a fast and fairly readable way, for example.

```{python}
import numpy as np

X = np.random.normal(size=(10, 5))
Y = np.random.normal(size=(2, 5))
sigma = 0.1

# calculate all pairwise distances between rows in X and Y
# (x - y)^2 = a^2 - 2 ab + b^2
dist_sq = np.square(X).sum(axis=1)[:, np.newaxis] - 2.0 * np.dot(X, Y.T) + np.square(Y).sum(axis=1)

# turn this into an RBF kernel (math might be off, but hopefully the point is clear)
rbf = np.exp(-np.sqrt(dist_sq) / (2 * sigma ** 2))

print(dist_sq)
print(rbf)
```

In R you have to hope someone vectorized the components of the operation that
you care about, copy an object along an appropriate dimension, or use
loops/applys/maps, which I think make the math harder to think about.