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What I try to do is find a transformation matrix that transforms camera's FOW into a real world plane (for now, assume the plane equation is z = 0). The camera is attached at point C(0,0,3) and the camera's direction is described by a vector (1,1,-1). The camera's field oview is 2.0 x 1.33 m at 2, angle of view 54 x 41 degrees. The origin point of the camera's FOW is it's left-top point O'(0,0). It looks similar to this picture (sorry, it was easier to draw it by hand) enter image description here

As far as I see it, the camera's field of view has to be rotated and moved. I've tried to use this formula to rotate the normalization vector about 45 degrees, but it seems that I only scaled the vector.

What is the matrix required to transform the camera FOW into a real world plane?

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    This does not appear to be a complete camera spect: a camera at $(0,0,2)$ looking in direction $(1, 1, -1)$ can be rotated about the view axis to any one of a whole circle of possible orientations. Are you perhaps assuming that the "up" direction on the film plane is as nearly vertical (in space) as possible? If so, which of your coordinates denotes "vertical" in space. (Both "y" and "z" are used for this in various domains.) – 2017-01-07
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    @JohnHughes I guess, it is "z" – 2017-01-07
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    @JohnHughes I've uploaded an image. Maybe that will make it more clear... – 2017-01-13
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    You still haven't resolved the last degree of freedom: in your figure, any rotation of the rectangle about the arrow still has the same field of view angles, same view vectors, same camera origin. Also, the notion of "the matrix required to transform the camera FOW (by which I expect you mean the rectangle you've drawn) into a real world place" is ambiguous: there are many $4 \times 3$ matrices (I'm using homogeneous coordinates) that accomplish this, for if $M$ is such a matrix, precomposing with a shear that leaves the rectangle plane invariant yies another such matrix. – 2017-01-13
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    A guess: Are you looking for the "world" coordinates of the point $A$ when you have the coordinates of $A'$ in a coordinate frame in which the point $O'$ has coordinates $(0,0)$? If so, as already implied in previous comments, we can guess where the tip of the arrow in the diagram is, but you haven't said how you would know where $O'$ is, nor even which edge of the rectangle is the $x'$ axis and which is $y'$. – 2017-01-13

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Hint: I strongly suggest you pick up any book on computer graphics (including the one I helped write, but others are equally good or better, I'm sure) or any online resource on the subject -- ACM SIGGRAPH has several good ones, I believe! -- and read the material about camera specifications and transformations. That will allow you to formulate your question properly, and more likely, will allow you to answer it for yourself. It's true that it's a lot of reading -- our chapters on these two topics probably occupy 40 pages -- but that's also why you should go look at a book: it's not reasonable to ask us to type 40 pages of material here.

In particular, the first edition of Foley and van Dam, Interactive Computer Graphics, should be available really cheap, and actually has formulas for stuff like this. A more modern treatment is Pete Shirley's book.