Lorentz transformation and rotation matrix

The rotation in the opposite direction (with $-theta$) is also a Lorentz transformation. Conventions about which one to use as an example have no physical significance.

This is similar to asking why we use a right-hand rule rather than a left-hand rule to define a cross product of two vectors. The answer has nothing to do with physical laws.

First of all, unless I've really misunderstood your question, this has very little to do with the Lorentz Transformations, this is just a rotation in four dimensional space, in the "$x_2x_3-$plane".

Your question seems to be about the sign of $theta$. Clearly, if you shift $theta$ to $-theta$ you would get the transformation you are "familiar" with. The two dimensional rotation matrix that represents counterclockwise rotations is $$R_{theta} = begin{pmatrix}costheta & -sintheta sintheta & costhetaend{pmatrix},$$

since it takes a vector $begin{pmatrix}x & yend{pmatrix}^text{T}$ to:

$$begin{pmatrix}xyend{pmatrix} ;longrightarrow begin{pmatrix}costheta & -sintheta sintheta & costhetaend{pmatrix}cdot begin{pmatrix}xyend{pmatrix} = begin{pmatrix}x costheta - y sintheta xsintheta + ycosthetaend{pmatrix}.$$

This is called an active rotation of vectors (in the counterclockwise direction, in a coordinate system that's right handed. i.e. one in which if $mathbf{hat{x}timeshat{y} = hat{z}}$.)

All of these assumptions are needed for the rotation matrix to be defined as it is, if any of them is changed, then the form of the matrix must change as well.

For example, instead of actually rotating the vector, you could alternatively rotate the entire coordinate system itself. It shouldn't be too hard to see that rotating a vector counterclockwise gives the same result as rotating the coordinate system clockwise (this is called a passive rotation). In this case, if you want to find how the basis vectors have changed, you need to use the rotation matrix with $theta to -theta$, which (happily) is just $(R_theta)^{-1}$.

you want to transformed the components of the vector $$vec v=begin{bmatrix} x' y' end{bmatrix}$$ from a rotating system to inertial system.

If you look at the figure the y component is:

$$y=sin(theta),x'+cos(theta),y'$$

and the x component is:

$$x=cos(theta),x'-sin(theta),y'$$

thus the transformation is

$$begin{bmatrix} x y end{bmatrix}=begin{bmatrix} cos(theta) & -sin(theta) sin(theta) & cos(theta) end{bmatrix},begin{bmatrix} x' y' end{bmatrix}$$

and

$$begin{bmatrix} x' y' end{bmatrix}=begin{bmatrix} cos(theta) & +sin(theta) -sin(theta) & cos(theta) end{bmatrix},begin{bmatrix} x y end{bmatrix}$$