"Linear algebra" essay: applying to the knowledge

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

Linear algebra is the combination of a group of numbers by linear combinations to get another set of numbers. The so-called linear combination, like the purchase of vegetables, I can buy half a catty of vegetables, two carrots and a broccoli-all the vegetables by a constant last added up to calculate a general ledger. I would never say I want to buy vegetables on a carrot, or broccoli divided by two pounds of vegetables. The multiplication between the dish and the dish is meaningless. So buying vegetables is a linear combination.

As I said before, I learned the linear algebra because I had to use it to write programs at work. This involves the problem in fact, everyone in high school often do this kind of topic, is the coordinate transformation. In the eyes of mathematicians, all geometric figures are numbers or formulas. The most basic of these is the point, and the point on the plane can be used two number: x-coordinate and the y-coordinate, the space of the point as long as a z-coordinate can be accurately expressed. Points can form a line, a constituent plane, a constituent body, to represent an object, as long as it gives enough coordinates of the key points.

The purpose of using numbers to represent points is to use the calculation of numbers to represent the relationship of points, and after summing up and analyzing, the motion of the rigid body is decomposed into a combination of displacement and rotation.

For example, Xiao Ming from home to the school line from home to start, can be decomposed into:

1.100 meters forward

2. Turn left 90 degrees

3.50 meters forward

4. Turn right 90 degrees

5.50 meters forward

6. Turn left 45 degrees

7. Reach the school 200 meters ahead

Fortunately, the transformation of coordinates can be expressed as a linear combination of coordinates. That is, from the original coordinates to the new coordinates just by multiplying the previous matrix.

The basic transformation matrices for two-dimensional coordinate displacements and rotations are readily available:

Displacement:

Rotating:

Why is the third-order matrix used to calculate two-dimensional coordinates? That is because the displacement transformation is to add a constant to each coordinate component, and in the 2-step matrix each item will eventually be multiplied by x or Y, and there will be no independent constant. The solution is simple, add one dimension to the two-dimensional coordinates, and become (x, Y, 1), so that one of the results in the multiplication will be multiplied by 1, allowing the constant term to exist. With a basic matrix, any complex transformation can be written as a series of matrices multiplied.

Now we look at Xiao Ming's Road to school, if Xiao Ming's home coordinates for the origin, facing the y-axis forward direction, the school coordinates can be expressed as:

Where O is the origin coordinate, fxxx is the forward matrix, Rxx is the right-turn matrix, LXX is the left-turn matrix.

To the three-dimensional space, the above conclusion is also established, but the matrix becomes 4 order, and the rotation matrix contains more information. Because different from the two-dimensional plane only the dimension one by one axis, the three-dimensional space can be any straight line axis, so the three-dimensional rotation matrix also contains the direction of the axis of the content.

The situation of the three-dimensional space is slightly more complicated. Lenovo Gymnastics and diving action commentary, but also forward and backward, but also swivel, this is actually the athletes around the rotation of different rotating shafts. Diving may not be a good example, because the body is flexible, so it is not a rigid body.

We can imagine a chopsticks also want to experience diving this sport, it is truly a rigid body, its choice of action is 3 weeks and a half, rotate 270 degrees, and then back to churn 2 weeks finally fell to the ground, then it landed on the head or feet toward the ground?

If the problem of the big brain hole is solved by the coordinate transformation, simply write the initial direction of the chopsticks into a matrix, and then multiply the three rotation matrices to get the answer immediately.

Coordinate transformation of three difficult spaces another important application is the computer three-dimensional graphics, which is also my work involved in the content and I learn linear algebra source power. All the things we see in 3D games are computer programs that display tens of thousands of basic graphics (triangles) on two-dimensional screens through coordinate transformations, none of which are implemented by multiplying matrices. You're right, coordinate transformation can also transform three-dimensional graphics into two-dimensional graphics. This transformation matrix is called the projection matrix, and its name and principle come from the realization of the world in the light to the object on the ground or wall shadow, which is called a positive projection. There is also a more commonly used projection method can be projected on the screen near large and small effect, called perspective projection.

In addition to the displacement, rotation and projection matrix, the computer also uses a scaling matrix, the three-dimensional graphics zoom out, so that the same thing in the game, the size of the difference does not need to do two copies. Triangles, together with four basic matrices, form the complex three-dimensional virtual world we see on the screen.

If this is the case, coordinates and matrices are good, purely physical, and will not arouse my interest in math books. What really appeals to me is the reverse process of this process. We've been able to accept it. The screen is two-dimensional, and the computer can easily be displayed on the screen simply by multiplying the three-dimensional graphics onto a matrix specified by the programmer. But things are not over, the user's mouse input is also two-dimensional, the mouse on the screen click on the calculation can get this screen point on the two-dimensional coordinates, but how to know that the two-dimensional point corresponding to the three-dimensional world which object? Do you want the programmer to specify a reverse matrix again? Does not seem to realize, because the programmer also does not know this matrix is what, even if knows, with our mentality also absolutely lazy to do. What to do? Go to school!

So there is this essay series, the previous one has been said, for any matrix multiplication:

can find an inverse matrix of M or a reciprocal, which makes:

Only the inverse matrix of the product of all forward matrices is required. Wow, Genius!

"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: applying to the knowledge