He studied for a long time, and finally to "hyperplane" had a preliminary understanding.
- n-dimensional space hyperplanes determined by the following equation:
Wherein, W and x are n-dimensional column vector, x is a point on the plane, w is the normal vector on the plane, hyperplane determined direction, b is a real number, representing the distance from the origin to the hyperplane. And
So, w Why is it normal vector? b Why is the distance the plane to the origin of it? Detailed explanation is given below:
- Our understanding of the concept of "plane" is generally defined in three-dimensional space, that is,
This plane is defined by two properties:
- 1, the equation is linear, is a linear combination of the components of the spatial points.
- 2, 1 is the number of equations. The plane is based on "three-dimensional" in.
If we put aside the "dimension" this limit, then there is a hyperplane defined. In fact, the hyperplane is purely mathematical concept, not a physical concept, it is a plane in a straight line, to promote the space plane, only when the dimension is greater than 3, it is called "super" plane . It's essentially a spatial dimension smaller than 1 degree of freedom.
Added:
the concept of freedom can be simply understood as the value to at least give the number of components to determine a set point . For example, (ultra) as long as the given (x, y, z) any two components, the remaining values of a plane in the three-dimensional space is determined. First determine the values of the two components is free, because they I think what values can take any value; the rest is "not free" because its value has been determined by another two-dimensional space of hyperplane is a straight line one-dimensional space hyperplane several axes. a point on.Baidu Encyclopedia of mathematical definition hyperplane is such that: hyperplane H from the mapping n-dimensional space to a subspace of n-1 dimensional space , which has a n-dimensional vector and a real definition. Because it is a subspace, so hyperplane through the origin certain .
Hyperplane explanation:
Typically, R2 set of points (two-dimensional space) i = (x, y) satisfies equation (i actual set point is a straight line):
AX +. 1 / C = 0 by + (. 1) (herein, 1 / b good showing for subsequent calculations)
wherein, a, b, c are scalar, a, 1 / b at least one is not 0. we assume that b is not 0. , Then
y = -abx - cb
At this time, transducer element method, so that t = x, (apparently, t is a scalar) of the point set i (x, y) can be expressed as
i (x, y) = ( t, -abt - cb) = t (1
, -ab) + (0, -cb) What is this line that? Actually over (0, -cb) point, direction (1, -ab) straight line L. Further, we make the vector n = (a, 1 / b ), then (1) can be expressed as a
n-I + C * = 0 (2)
Magic moment arrived. Is assumed to take that p0 (x0, y0) on the straight line L, obviously, n * p0 + c = 0 , then c = -n * p0. Still further, the (2) can be rewritten to give n * in * p0 = 0, can be n-* (I - P0) = 0 .
Since n and (i - p0) are vectors, (i - p0) on the straight line L, so, n vertical line L, i.e., the straight line L n is a normal vector. Furthermore, we can obtain, with the points and the difference vector p n orthogonal vectors is the set of points i (x, y).
What is further explained hyperplane:
Given a vector space Rn, a point P and a nonzero vector n, satisfying
n * (i - p) =
0 is said to point set by the point p i is the hyperplane by the normal vector n is a vector hyperplane. According to this definition, although when a dimension greater than 3 can become "super" plane, but you can still believe that a straight line is a hyperplane in R2 space plane is a hyperplane in the space R3. Hyperplane n in Rn is a space in Rn - 1 dimensional affine subspace.
Distance hyperplane point:
Personal summary:
Read a lot of blog and some video, hyper-plane is nothing more than two content, it is a concept, a point is to calculate the distance to super plane. We can put super-dimensional plane is seen as the only change have become more common plane, the other nothing special. In addition, we can use mathematical formulas to represent it, but did not give it the right way in the sense of space, which is many teachers taught in the area and the places.
Only the development of mathematical physics ahead of the development of our society will progress. In mathematics, we can calculate the length of the n-dimensional vector (or norm || w ||), but we did not express its sense of space, but we currently have no way to express.
Just a few more abstract mathematics, do not worry, step by step, and then a long way to come to an end! Refuel
Reference link:
. 1, https://blog.csdn.net/dengheCSDN/article/details/77313758
2, https://www.jianshu.com/p/2dadd6f8cdbd
. 3, http://www.sohu.com/a / 206572358_160850
