Data Science in high dimensions-linear space (upper)

The rule f that causes the elements of the set Y to correspond to the elements of the collection X.

The concept of generalized:

Movie tickets are also a kind of mapping, pay is also a mapping, male and female friends are also mapping. As long as there is a correspondence, I can think of it as a mapping. The concept of mapping is an abstraction used to describe the relationship between nature and society.

It is important to remember: the concept of mapping is a very broad concept, any two related things can be described by the concept of mapping, such as Zhang San mapping to 31 classes, high-latitude vector mapping to low-dimensional space.

The concept of mapping and linear space is crucial for data science, as real-world data always contains many dimensions. So the mathematical tool of linear space is, in a way, a perfect match for data science.

Like

X is mapped to Y on the resulting Y value is the X-map becomes the image

For example, X is the class boy, and one of the boys ' girlfriends is his image.

Range and Definition fields

The value of x is defined as the domain

X the value of the image on Y as a range of domains

Single shot and full shot

A case study of male-female relationship mapping

1. When the mapped range is equal to Y, the map is called full-shot

That all the girls have boyfriends, then they're full-shot.

2. Different x cannot be mapped to the same Y, this mapping is called a single shot

If there's a girl with two boats on her feet and two or three boyfriends, there's no doubt it's not a single shot.

3. When the map F is a single shot and full shot, we call the mapping one by one map or double shot

That all the boys have only one girlfriend, all the girls have only one boyfriend

Linear mapping

from the vector.

For any vector in plane R (2), the definition of addition and multiplication is already clear, and the two operations are closed, that is, the result of the operation is still in R (2), and the two operations satisfy the following eight operations.

The concept of linear space

By the knowledge of linear algebra, two-dimensional space can be generalized to n-dimensional space, so we push the above situation to n-dimensional real vector space (note the difference between vector space and linear space), we can know that the vectors in the n-dimensional vector space are also satisfied with some of the above operation rate.

Because of the universal nature of real vector space, we want to define a special name for it. Then there is the definition of linear space. Note: But the concept of linear space is larger than the real vector space.

linear Space Definition : If the elements of a non-empty set, the addition and multiplication of two operations is closed , and its elements of the operation of the law satisfies the above eight rules of operation, Then we call the set of linear space or vector space on the domain F (Number field is the most basic element in the collection).

The basic properties of linear space

Base (Basis), coordinates (coordinate), and dimensions of linear space (Dimension)

base : If a set of vectors in a linear space is linearly independent, and any vector in a linear space can be represented as a linear combination of this set of vectors, then this group of vectors is called the base of a linear space.

The definition of radical shows that there are countless bases in a linear space-any one base, as long as a constant quantity of its interior is multiplied by the constants to become a new vector, instead of itself, the new set of vectors is still the base of the linear space.

coordinates: any vector in a linear space can be represented by a linear combination of the base vector, and the parameters of the linear combination are the coordinates of the vector relative to the substrate.

Dimension: the number of vectors in the base, which is the dimension of the linear space.

Note: The dimension of a vector in linear space is not necessarily a dimension of linear space . As in, the linear space consisting of X1 and X2 is only two-dimensional, but X1 and X2 are 3-dimensional vectors. The dimension of a linear space, depending on the number of vectors contained in any of its bases.

Some notes on the basis of linear space

1. If the linear space is regarded as a vector group of countless vectors, then the base of the linear space is the largest independent group of the Vector group, and the dimension of the linear space is the order of the vector group. (for example, the rank of a vector group consisting of all vectors of the linear space spanned by the x1,x2 two vectors, no doubt 2)

2. If the a1,a2...an is a base of a linear space V, the linear space can be written as Span{a1,a2,... an}

3. For any linear independent group in a linear space, if the number is equal to the base, then the Linear independent group is also a base

4. Studying n-dimensional vector space V, which is represented by its base and vector coordinates , is transformed into a linear space r^n

The default base

In fact, in linear space r^n, our daily 3-dimensional coordinates have a default base, which is the unit matrix of N*n. Be sure to remember that coordinates and vectors are not the same thing.

SubSpace of linear space

Definition: 1. The subspace is a subset of the linear space, and the dimensionality of the base of the subspace is smaller than the number of the base of the parent space; 2. SubSpace must contain the origin point in order to satisfy the closeness of the addition and multiplication operations and the 8 Operation rules.

Base transformation and coordinate transformation

For different bases, the coordinates of the vectors on the linear space are of course not the same, so how should the two coordinates be transformed?

Base transformation

Coordinate transformation

The proof is simple, multiply the base vector group with the coordinates, the different bases and their corresponding coordinates multiply, the two bases and the coordinate multiplication result is same. Is the value of the vector.

Data Science in high dimensions-linear space (upper)