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Why is linear algebra actually useful? There very many applications of linear algebra. In data science, in particular, there are several ones of high importance. Some are easy to grasp, others not just yet. In this lesson, we will explore 3 of them:

• Vectorized code also known as array programming
• Image recognition
• Dimensionality reduction

Okay. Let’s start from the simplest and probably the most commonly used one – vectorized
code. We can certainly claim that the price of a house depends on its size. Suppose you know
that the exact relationship for some neighborhood is given by the equation:

Price equals 10,190 + 223 times size. Moreover, you know the sizes of 5 houses 693, 656, 1060, 487, and 1275 square feet.

What you want to do is plug-in each size in the equation and find the price of each house,
right?

Well, for the first one we get: 10190 + 223 times 693 equals 164,729. Then we can find the next one, and so on, until we find all prices.

Now, if we have 100 houses, doing that by hand would be quite tedious, wouldn’t it?

One way to deal with that problem is by creating a loop. You can iterate over the sizes, multiplying
each of them by 223, and adding 10,190. However, we are smarter than that, aren’t we? We know some linear algebra already. Let’s explore these two objects:

A 5 by 2 matrix and a vector of length 2. The matrix contains a column of 1s and another – with the sizes of the houses. The vector contains 10,190 and 223 – the numbers from the equation.
If we go about multiplying them, we will get a vector of length 5. The first element will be equal to:

1 times 10,190 plus 693 times 223. The second to:
1 times 10,190 plus 656 times 223. And so on.


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