space

In a broad sense, we define everything through the universe, which is also the unity of space and time. For the whole physical world, time and space are the two most important and essential dimensions. If time is removed, things can be described in terms of space. Space can accommodate things, just like people and buildings in the real world are all in a certain space.

Mathematics space

The concept of space can also be extended to other fields, such as mathematics where space is a collection of points and geometric structures. The most famous is our daily life most contact is Euclidean space, corresponding to the ancient Greek mathematician Euclid created Euclidean geometry. Compared with low dimensions, there are mainly plane geometric space and solid geometric space, in which a series of concepts such as distance, Angle and inner product are defined and related constraints are stipulated.

If two and three dimensions are generalized to finite N-dimensions, all defined Spaces from two dimensions to finite N-dimensions are collectively referred to as Euclidean Spaces. So what are the main definition constraints? Euclidean space mainly has five constraints: satisfy the constraint of distance, satisfy the constraint of linear structure, satisfy the constraint of norm, satisfy the constraint of inner product, must be finite dimension.

Vector space

The object corresponding to vector space is vector. After introducing the concept of vector, the treatment of many problems will become more concise and clear.

We can intuitively feel the vector space is generally two-dimensional and three-dimensional vector space, that is, corresponding to the plane coordinate system (X-axis and Y-axis) and three-dimensional coordinates (X-axis, Y-axis and z-axis). But actually, in addition to two and three dimensions, vector Spaces can also be generalized to finite n-dimensional vector Spaces. The important thing about vector space is that constraints are linear constraints, that is, you can add, you can multiply, you can do commutative, associative, distributive, so vector space is also called a linear space.

Vector representation

A vector is a quantity that has magnitude and direction. It can be represented by an arrow. The length of the arrow indicates the magnitude of the vector, and the direction of the arrow indicates the direction of the vector. In physics, vectors are used as the equivalent of vectors, while in computing, arrays or lists are used to represent vectors. In the figure below, the central point (0,0,0) and point P (2,3,5) together determine a vector, which can be expressed as:

Vector abstraction

The mathematical definition of a vector is abstract, so what does it do? From a higher level, vector is a kind of abstract thinking of things, but also a very useful tool, converting things into vector system can solve many problems efficiently and succinct. We can map things to vectors, or we can map features of things to vector Spaces.

In fact, anything or a feature can be abstracted as a vector. Representation of things as vectors is the first step in model processing, and once we have abstracted things into vectors we can then model and process them.

matrix

A matrix is a rectangular array composed of m rows and n columns. In contrast to a vector, a matrix can actually be regarded as an object composed of a group of vectors. For example, the previous word vector is a special matrix of m rows and 1 column, so if the specified number of n words form a batch, then it is m rows and n columns matrix.

For vector Spaces, the essence of a matrix is to apply transformation operations to vectors, that is to say, matrices describe transformations. For example, the following expression, vector x is transformed into vector y by the transformation described by matrix A.

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