Reference:
https://baike.baidu.com/item/%E8%B5%8B%E8%8C%83%E7%A9%BA%E9%97%B4/2285667?fr=aladdin
http://blog. csdn.net/mr_hai_cn/article/details/53207307#reply
http://blog.csdn.net/soudog/article/details/2050632
https://www.zhihu.com/question/19967778
distance
First we give the definition of distance. to understand the mathematical definition of distance abstraction.
If the above 3 conditions are met, we call d(x,y) the distance between these two points.
In addition, we need to understand that distance measures a kind of length. Since distance is the quantification of length, it may be a straight line or a curve. Just like finding the distance between 2 points on a globe, you need to draw a big circle. , then find the arc length.
distance between vectors
The distance discussed above is only a feature of distance that is highly generalized and abstract. Here are some commonly used specific distance formulas between vectors. Case 1 to case 3 are Euclidean distance, chessboard distance, Manhattan distance (also called city distance).
vector norm
The vector norm can be seen as the distance from the vector x=(x1,x2,...,xn) to zero.
When we understand the norm in this way, then for the distance formulas in the above three cases, the norm of the corresponding vector we get is:
We will find that the norm of a vector is very similar to the formula for the distance between vectors, that is, The case where the y vector is 0. The reason why all vector norms are based on the 0 vector is because the 0 vector is the only constant in the linear space.
relationship of various spaces
Before we continue our discussion, I would like to give a general conclusion to give you a general understanding of the interrelationships of various spaces proposed in mathematics. Linear space (vector space) is a relatively elementary space. If a norm is defined in it, it becomes a normed linear space. If the normed linear space satisfies completeness, it becomes a Banach space; if an angle is defined in the normed linear space, there is an inner product space, and if the inner product space satisfies completeness, a Hilbert space is obtained. These spaces are all linear spaces .
So what is completeness?
What if we want to study convergence (limit)? —Definition completeness
Simply put, the space is in the limit operation, and the limit cannot be run out. Therefore, it is obvious that the set of rational numbers and the set of irrational numbers are not complete. The set of real numbers is complete.
Simplified presentation of the differences between spaces
1. Linear space (vector space)
Linear space is also called vector space, focusing on the position of the vector. For a linear space, knowing the basis (equivalent to the coordinate system in the three-dimensional space) can determine the coordinates of the elements in the space (ie position); linear spaces only define addition and multiplication operations.
What if we want to know the length of the vector? ——Define norm, introduce normative linear space
2. Normative linear space defines the linear space of norm! !
What if we want to know the angle of a vector? ——Define the inner product and introduce the inner product space
3. The inner product space defines the linear space of the inner product! !
4. Euclidean space defines a finite-dimensional real linear space for inner products! !
What if we want to study convergence (limit)? ---Definition complete
5.Banach space complete normed linear space! ! !
6. Hilbert space complete inner product space! ! ! (The limit operation cannot run out of the measurement range)
linear space
Linear space, also known as vector space, focuses on the position of the vector. For a linear space, knowing the basis (equivalent to the coordinate system in three-dimensional space) can determine the coordinates (ie position) of elements in the space; linear space only defines Addition and multiplication operations.
What if we want to know the length of the vector? —Define norm and introduce normed linear space. That is, in linear space, we cannot find the length of the vector.
Subtle Differences Between Vector Spaces and Linear Spaces
Linear space and vector space are basically the same thing, but the elements in the linear space can be anything; after the basis is selected, it can be expressed in the form of a vector, so the linear space is also called a vector space.
Specifically, the vector space is narrow, and its elements can only be vectors. Linear space is generalized, and its elements can be anything, including vectors, matrices, polynomials, functions... After the linear space has a basis selected, it can be expressed in the form of a vector. At this time, the two have the same meaning.
Some intuitive understanding of linear space
Excerpted from http://blog.csdn.net/soudog/article/details/2050632 , thanks to the author.
In linear space, when you choose a set of bases, you can not only use a vector to describe any object in the space, but also use a matrix to describe any movement (transformation) in the space. The way to make a certain object move correspondingly is to multiply the matrix representing that movement by the vector representing that object.
In short, after a basis has been chosen in linear space, the vector characterizes the object, the matrix characterizes the motion of the object, and the motion is imposed by multiplying the matrix by the vector. Or we can also understand a matrix as a mapping, a transformation function that maps the original point to the end point of the desired mapping through matrix multiplication.
It should be noted that the motion in the above sentence does not mean motion in the true sense, that is, the process of its result (forming the end point of the mapping) does not pass through any irrelevant point in this linear space, That is, no path is formed to reach the end point. Therefore, the matrix depicts the motion of the object, and it is more accurate to use the multiplication of the matrix and the vector to apply the motion and change the matrix to the description of the transition in the linear space.
base
A basis is a set of linearly independent vectors whose inner product is 0.
Intuitively, base selection is to find a coordinate system.
similarity matrix
If the matrices A and B are two different descriptions of the same linear transformation (the difference is because different bases are selected, that is, different coordinate systems are selected), then a non-singular matrix must be found. P, so that A and B satisfy the relationship:
the above formula is also the definition formula of the similarity matrix.
From this point of view, the so-called similarity matrix is the description matrix of the same linear transformation on different bases. That is to say, a family of similar matrices are all descriptions of the same linear transformation under different bases.
Normed space (normed linear space)
First, the definition of normed space on Baidu Encyclopedia is given.
As can be seen from the above definition, the concept of norm is added to the normed space . Therefore , the linear space with the added norm is formed into a new space called the normed space . Therefore, the normed space has the modulo length of the vector, that is, the norm. That is, length can be discussed . But there is still a lack of a very important concept - the angle between two vectors. In order to overcome this defect, we introduce the concept of inner product.
inner product space
An inner product space is a vector space with an extra structure added. This additional structure is called the inner product, or scalar product, or dot product. This added structure allows us to discuss strictly the "angles" and "lengths" of vectors .