Picture fuzzy sub-hyperspace of a hyper vector space and its application in decision making problem

In this paper, the notion of picture fuzzy sub-hyperspace of a hyper vector space is
introduced and some related results are investigated on the basis of some basic operations (intersection,
union, Cartesian product etc.) on picture fuzzy sets. The concept of picture fuzzy linear transformation
with respect to some picture fuzzy sub-hyperspace is initiated here and some important results are
studied in this regard. It is shown that with respect to some pre-assumed picture fuzzy sub-hyperspace,
linear combination of two picture fuzzy linear transformations is a picture fuzzy linear transformation,
composition of two picture fuzzy linear transformations is a picture fuzzy linear transformation and
inverse of a bijective picture fuzzy linear transformation is a picture fuzzy linear transformation. The
effect of good linear transformation on picture fuzzy sub-hyperspaces is discussed here. It is shown
that the image of a picture fuzzy sub-hyperspace is a picture fuzzy sub-hyperspace under bijective good
linear transformation and the inverse image of a picture fuzzy sub-hyperspace is a picture fuzzy subhyperspace under good linear transformation. Some important results on picture fuzzy sub-hyperspaces
in the light of (θ, φ, ψ)-cut of picture fuzzy set are studied here. Finally, an application of picture fuzzy
sub-hyperspace conditions in decision making problem is presented here