In this article, an attempt is made to find all linear transformations that map a standard Pythagorean triple (a Pythagorean triple [x y z][superscript T] with y being even) into a standard Pythagorean triple, which have [3 4 5][superscript T] as their fixed point. All such transformations form a monoid S* under matrix product. It is found that S* finite with only five (5) elements including the identity transformation. It is interesting to know that if the same finiteness holds when we replace [3 4 5][superscript T] by other standard Pythagorean triples. The results of this analysis will be of interest to a broad spectrum of readers. It bridges the concepts such as fixed point, linear transformation, and Pythagorean triple. It should be of pedagogic as well as theoretical value.