"Linear algebra" essay: 've Seen

• To pick up the flowers
• Don't forget beginner's mind
• Apply
• 've seen
• Bit by bit

In general, we measure whether the two points are close to the standard by the distance between two points. The distance of two points on a ruler is simply subtracted from the value of the scale on which the two points are located. If the distance between two points on a two-dimensional plane, say a diagonal distance of a square with a side length of 1, we will use the Pythagorean theorem:

Now further to the three-dimensional space on the distance of two points, such as an edge length of 1 cube diagonal distance, you can extend the Pythagorean theorem:

The distance from the geometry ends here, and algebra is very modest in learning the concept from geometry. For a vector coordinate x= (x1, x2, x3 ... xn), the inner product operation is defined:

The result is the square of the distance from the point to the origin of the coordinates. But in the algebraic world there is no limit to the three dimensions of the geometry world, vectors can represent the coordinates of any dimension, and the inner product can also be applied naturally to any dimension of space.
With the concept of distance, of course we have to do something interesting with it. or from the concept of geometry, we can use distance to calculate the perimeter and area of a plane graph and the volume of the spatial graph. For example, we can calculate the coordinates of a point in a line or a plane that is closest to the nearest point on the straight or flat plane. The two applications in mathematics, the former produced a branch of calculus , the latter is included in the study of linear algebra scope.
To return to geometry, to find the nearest point in a line or plane, usually we need to make a perpendicular line between this point and the straight or the plane, and its perpendicular is the point we require. This point has a larger name, called the projection of the point on a line or plane .
In linear algebra, a straight line passes through the origin, and the coordinates of any point on the line represent the vector from the origin to the point, and this line is the linear combination of the vector itself, that is, the point in which the vector is enlarged or reduced by any number of times is the straight line. To simplify a little bit, we select the unit vector with a distance of 1 from the Origin on this line, and the magnification or reduction is equal to the distance from the point to the origin, namely:

The mathematician tells us that the distance from the origin of the projection from X to the line y is the inner product of X and E. Wow, that's so easy! This lazy thief is creative and has a level. But do you think that the projection on a straight line is done? Mathematicians don't think that way. They further tell us that for a plane that is over the origin, the plane can be represented by a linear combination of two non-collinear vectors. To simplify the words, it is also selected two and the origin is a distance of 1 of the unit vector, while the two vectors are perpendicular to each other, they give the unit vector a name, called "base"(yes, "base" is an important work in linear algebra). Then the plane can be represented as: then the projection of any point x to the plane y is: the mathematician is out of order, and they infer the arbitrary dimensional vector space according to the line and plane: what? More than three-dimensional space humans have yet to find? I don't care, anyway, my vector wants to be a lot of dimensions. I also know that at any point in this space, the projection of x to y is:

"You believe, or you do not believe, it is there." does not increase or decrease. ”

The distance in linear algebra is far better than that of geometrical space because of its geometry. The use of it as a fanciful, but often have unexpected results.
We can already find a point in the line closest to the point. What if it's the opposite?

Are you familiar with this situation? Yes, we often meet from the beginning of Junior High School physics experiment, just don't realize how big the thing we are doing.
In the physical experiment class for measuring spring elasticity coefficients, we were asked to measure the tensile force on a set of tension gauges with different lengths of spring stretching. Then, according to Hooke's theorem "F=kx", the elastic coefficient K of spring is obtained.
So Xiao Ming in accordance with the requirements of the physics teacher earnestly and carefully stretching the spring, record length and relay. In his efforts to get 5 sets of experimental data:

The experiment must have the error, each group of data calculated K is different also very logical. Xiao Ming put these 5 k to an average of 2.84, intends to give the physics teacher to go. But with Xiao Ming in the same group of red but with another method of calculation: She took five sets of experiments with x as the horizontal axis, F for the ordinate on the grid paper five points, and then very carefully drew a straight line, looks close to each point. She calculated the slope of the line: 2.95. Finally, the physics teacher praised the small red, because the elastic coefficient of the spring is 3.
Xiao Ming and Xiao Hong, although using different methods, but they have found that they think with the five points closest to the line. Xiaoming's method has an accurate answer, but this is the simplest method of mathematical averaging, using a single equation. While the method of little Red does not have a formula, but she measured is true geometric distance, if written equation is two times, this is very consistent with the spirit of the law of the hook, but she in the recent selection of straight line when there is a subjective nature. The final result is that although Xiao Hong is closer to the right result than Xiaoming, the two-person approach is not the most accurate.
Of course, the methods of Xiao Hong and Xiao Ming are sufficient in daily life, although their methods are only applicable to specific problems. But this is far from enough for mathematicians, who are pursuing the single most accurate value in theory, and are universally applicable to all problems. In the end they locked a method by which the average of the two-second equation can be calculated, which is the least squares.
Xiao Ming's experimental data we can write a matrix equation:

Obviously, no matter how much k equals, this equation cannot be equal on either side. Mathematicians thought of how to make the two sides equal, a matrix equation:

Where A and B are constant matrices and X is unknown, we call this equation incompatible if we know that the equals sign will never be true. For incompatible equations, an x can be obtained to make:

To make the distance between B ' and B nearest, the method that requires such x is the solution equation:

If both A and B are vectors with only one column, the equation can be simplified to a vector inner product, i.e.:

By this formula to calculate the data of xiaoming:

The results are more accurate, but more importantly, the least squares can be applied to the statistics of any formula. The next section will start with the least squares and look at the most unknown "Fourier transform" in communication.

"Linear Algebra" series of essays started in the public number: the Cat library, which is sponsored by the director of the young bourgeoisie youth gathering, welcome attention.

"Linear algebra" essay: 've Seen