Left multiply right multiply matrix problem

When transforming a vector , The left and right multiplication matrices are actually the same （ Maybe someone will raise an objection ）, Our goal is to transform vectors , Vectors are interpreted the same by column or row .

a It's a vector ,M It's a matrix , The end result is a vector , The result of a vector and its transpose is the same , Only when multiplied by a matrix ：

According to the theorem of matrix transpose
（aM） The transpose = M The transpose * a The transpose

A key conclusion ：
aM = M The transpose * a The transpose
The row vector times the matrix is equal to the transpose of the matrix times the column vector （ Transpose of row vectors ）

This explains the different rotation matrices in some books

as follows ：
1.2D Rotation matrix
If it's a vector left multiply matrix , The rotation matrix is ：M
cos(angle) sin(angle)
-sin(angle) cos(angle)

vector a=(x,y)

Vector left multiply matrix ： a* M

And some books 2D The rotation matrix is like this ：M2
cos(angle) -sin(angle)
sin(angle) cos(angle)

Vector right multiply matrix ：M2 * a

This is the conclusion above ,a*M=M2 * a

The row vector times the matrix is equal to the transpose of the matrix times the column vector （ Transpose of row vectors ）

So some books use column vector right multiplication , Some are left multiplication of row vectors , To make a difference , The result is the same .

The row vector times the matrix , Each row of the matrix is the transformed basis vector .
Matrix Times column vector , Each column of a matrix is a transformed basis vector .

notes ： And the basis vector here is not normalized , It can be scaled differently .

Unity In the essentials of introduction, the object is transformed from object space to world space , Get the position and direction of the coordinate axis of the object in world space , Directly put the base vector of the coordinate axis into the first three columns of the matrix , When the position of the coordinate axis is placed in the fourth column, the matrix from the object to the world is constructed .

Similarly, it can be extended to 3x3、4x4 matrix

such as 4x4 Translation matrix , Some books put the offset in the fourth column , Some are put on the fourth line , The principle of same .