3. GEOMETRIC INTERPRETATION OF MATRICES

First, we can understand the matrix through vectors. Vectors [1,-3, 4] can be interpreted as the addition of the following vectors.

Arbitrary VectorVCan be written as the following extension form

Further written:

The unit on the right is X, Y, and Z axis.NX, NY, NZ. We can write it:

V = x *NX+ Y *NY+ Z *NZ

 

If we use a vectorP, G, RRewriteNX, NY, NZMeaning unchanged:

V= XP+ YG+ ZR

HereP, G, RThese are called base vectors. Here they are Cartesian coordinate axes. In fact, a coordinate system can be defined using any three linear-independent vectors as base vectors. [x, y, z] is a vector.VInP, G, RRepresents in the coordinate system of the base vector. ToP, G, RDefine a 3*3 matrix for rowsM, Get

Multiply the vector [x, y, z] by the matrix to obtain

We found it andVSo we get the following conclusion:

 

The row of the matrix is the base vector of the coordinate system,Multiplying a vector by this matrix is equivalent to executing a coordinate transformation. If am = B, it means that vector A under the original coordinate system is converted to vector B under the action of M (the specific function is coordinate transformation, and the base coordinate of the new coordinate system is p, q. (IfP, Q, RIf andNX, NY, NZIf the value is the same, it indicates that the coordinate system is still the original coordinate system and no operation is performed)

 

Multiply the matrix m by three base vectors, and we will find that:Each row of the matrix can be interpreted as the converted base vector..

 

The following example

 

Cartesian coordinate system VectorAIn the matrixM(The specific function is coordinate transformation. The base coordinate of the new coordinate system is p, q ).B. (BThe specific coordinate values are also represented by Cartesian coordinates. Cartesian coordinates are equivalent to the world coordinate system, which is the root coordinate system and the absolute position in the world coordinate system, however, vector B is generated after vector A undergoes M Transformation (including rotation and scaling .)

 

There is a painting in the original coordinate system.M= [2 1;-1 2] After transformation, the result is the image after rotation and stretching:

AM = B, WhereAIs the Original Coordinate (Cartesian coordinate system). The position of the vector relative to the coordinate system is shown in the left figure above,MThe matrix contains the base of the new coordinate system,MIt is used for rotation, scaling, and projection,BYesAInMNew vectors after the operation,

AThe relative position of the original coordinate system.MAfter the operation, the vector in the new coordinate system should be the same as the relative position of the coordinate system, that is (AIt is equivalent to the current location of x-y in the original coordinate system (BThe relative position of the transformed coordinate system p-Q ).AAfter the vector is rotated and scaled, a new vector is obtained.B, BThe specific location is represented by absolute coordinates (World coordinates), that is, Cartesian coordinates.

 

For example, the vector in the left graph aboveAThe relative position in the Cartesian coordinate system is determined (this relative position is not necessarily an angle. If it is only a rotation, the angle will not change. If it is scaled, the angle will change, after rotating and scaling, the new coordinate system is P-Q, and the new vector is obtained.BThe relative position of the new coordinate system is similar to that of the west of the Cartesian coordinate system ).MBefore transformation,AIs the diagonal line,MAfter conversion,BOr diagonal lines.

 

 

AM = BIt can also be understood that there is an object in which an object coordinate system is established. A point in an object connects to the origin of an object coordinate system to form a vector.A,MOperations on this object (rotation, scaling, projection, etc ),MAfter the operation, the orientation of the object coordinate system to the object remains unchanged (just like the human's front and back left, no matter what angle it is, my front and back are unchanged relative to my direction), VectorBA point in an object is a vector connected to the origin of the object at the new position A1 after the object is rotated. However,AAndBAll use world coordinates to describe absolute positions.

 

Summary: geometric meaning of Matrices

AM = B, where M = [P; q; R]

A is the vector in the world coordinate system. M transformation is performed on a. After M transformation, the new coordinate system uses P, Q, and r as the basis of the coordinate system, vector B is the value under the new base, andThe coefficients of P, Q, and R are the three coefficients of A respectively.. The position of vector B is described by the world coordinates (absolute coordinates.