[Questions about Machine Learning] Chapter I Mathematics Fundamentals

In this chapter, we will discuss some basic mathematics knowledge that you need to know for further study.

Q. What are the relations between scalar, vector, matrix, and tensor?

A. A vector is an ordered finite list of numbers. Vectors are usually written as vertical arrays, surrounded by square or curved brackets, as seen below.

$$\begin{pmatrix}-1.1 \\ 0.0 \\ 3.6 \\ -7.2 \end{pmatrix} or \begin{bmatrix}-1.1 \\ 0.0 \\ 3.6 \\ -7.2 \end{bmatrix}$$

Sometimes, they are written as numbers separated by commas and surrounded by parentheses. As seen below.

$$(-1.1, 0.0, 3.6, -7.2)$$

Vector is often denoted by a lowercase symbol $a$. We can get the element (also known as entries, coefficients or components) of a vector by the index, and the $i$th element of the vector $a$ is therefore denoted as $a_i$ where the subscript $i$ is an integer index of the vector. (Obviously, $0<i<n$).

If two vectors have the same size, and more importantly, each of the corresponding entries is the same, then the two vectors are equal, which is denoted as $a=b$.

A scalar is a number or a value. In most applications, scalars are real numbers. We usually use an italic lowercase symbol to denote a scalar. For example, $\textit{a}$ is a scalar.

A matrix is a rectangular array, which means it is a 2-dimensional data table at the same time. Matrix is a collection of items that have the same feature and character. In a matrix, a column indicates a feature, and a row indicates an item. Matrix is usually denoted as a capital letter, $A$ for example.

A tensor is an array with more than 2 dimensions. Generally, if the elements of an array are distributed in a regular grid with several dimensions, we would call it a tensor. We use a capital letter to denote a tensor, same with the matrix. $A$ for example. An element in a tensor is denoted as $A_(i,j,k)$.

Relations between them

Scalar is a 0-dimensional tensor. Vector is a 1-dimensional tensor. For example, with a scalar, we could get the length of a rod, but we cannot know the direction of this rod.

With a vector, we could know both the length and direction of a rod.

With a tensor, we may be able to know both the length and direction of a rod, and we could even know more about the rod. (for example, the degree of deflection)

Q. What are the differences between tensor and matrix?

From the aspect of algebra, the matrix is a generation of the vector, the matrix is a 2-dimensional table. $n$-dimensional is a so-called $n$-dimensional table. Noted that this is not a strict definition of the tensor.

For the aspect of geometry, a matrix is a geometric sense value. It does not change with the coordinate transformation of the frame of reference. the vector has this feature too.

The tensor can be represented by a $3$x$3$ matrix or an $n$x$n$ matrix.

A scalar can be regarded as a $1$x$1$ matrix while a vector with $n$ items can be regarded as $1$x$n$ matrix.

Q. What will happen if I multiply a matrix and a tensor?

A. You can only multiply an $m$x$n$ matrix and a $n$ items vector. Then you will get a $m$ items vector. The key to this is regarded each row of the matrix as a vector, and multiply the given vector.

For example, If you are going to multiply the following:

$$\begin{bmatrix}1, 2 \\ 0.0, 1 \\ 3.6, 3 \\ -7.2,2 \end{bmatrix}$$ and $$\begin{bmatrix}-1.1 \\ 0.0 \\ 3.6 \\ -7.2 \end{bmatrix}$$

Q. What is the norm?

In mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space. There are many different types of norms for a vector or a matrix. For example,

1-norm: $ ||x|| 1 = \sum{i=1}^N |x _i| $

2-norm or Euclid norm: $ ||x|| _2 = $


Q. What are the norms of matrix and vector?

We define a vector as $\vec{x}=(x1,x2,...,x_N)$. Its norm will be:

Q. What is the positive definite matrix?

Q. How to judge if a matrix is the positive definite matrix?

Q. What is a derivative?

Q. How to calculate the derivatives?

Q. What are the differences between derivatives and partial derivatives?

Q. What is eigenvalue? What is eigenvector? What is eigenvalue decomposition?

Q. What is the singular value? What is singular value decomposition?

Q. What are the differences between singular value and eigenvalue? and what about their decomposition?

Q. What is the probability?

Q. What are the differences between variable and random variable?

Q. What are the common probability distribution?

Q. What is the conditional probability?

Q. What is joint distribution? What is marginal distribution? What are their relations?

Q. What is the chain rule for conditional probability?

Q. What is independence and conditional independence?

Q. What is the expectation? What is variance? What is covariance? What is the correlation coefficient?