As I mentioned, neural networks are essentially functions, which are trained using the tools of calculus. However, they

Although vector spaces are infinite (in our case), you can find a finite set of vectors that can be used to express all vectors in the space. For example, on the plane, we have

When the inner product for two vectors is zero, we say that the vectors are orthogonal to each other. (Try to come up with some concrete examples on the plane to understand the concept deeper.)

Don’t panic! I know that this looks very scary (at least it looked that to me when I was a freshman student with a math major), but it really isn’t. These just guarantee that vectors can be added together and scaled just as you would expect. When thinking about vector spaces, it helps if you mentally model them as

where A is a matrix, b and x are vectors, and σ is the sigmoid function. (Or any activation function, really.) Well, the part Ax is a linear transformation. In general, the function

If you feel like you have a good understanding of vector spaces, the next step would be to understand how to measure a vector’s magnitude. By default, a vector space in itself gives no tools for this. How would you do this on the plane? You have probably learned that there, we have

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This is the prototypical model of a vector space. In general, a set of vectors V is a vector space over the real numbers if you can add vectors together and multiply a vector with a real number, such that the following properties hold:

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A basis always exists for any vector space. (It may not be a finite set, but that shouldn’t concern us now.) Without a doubt, a basis simplifies things greatly when talking about linear spaces.

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When the vectors in a basis are orthogonal to each other, we call it an orthogonal basis. If each basis vector’s norm is 1 for an orthogonal basis, we say it is orthonormal.

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ons and most hated by many people is documentation. In this context, a documented API is essential so that our users who consume it can understand several important aspects of our interface, including accessibility, responses, requests, examples.

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To have a good understanding of linear algebra, I would suggest starting with vector spaces. It is better if we talk about a special case first. You can think of each point in the plane as a tuple

One of the key objects related to vector spaces are linear transformations. If you have seen a neural network before, you know that one of the fundamental building blocks are layers of the form