Primarily we examine the new example of quasilinear spaces, namely, "IR" interval space." We obtain some new theorems and results related to this new quasilinear space. After giving some new notions of quasilinear dependence-independence and basis on quasilinear functional analysis, we obtain some results on IR" interval space related to these concepts. Secondly, we present Is, I-c0, Il(00) and Il(2) quasilinear spaces and we research some algebraic properties of these spaces. We obtain some new results and provide an important contribution to the improvement of quasilinear functional analysis.