The challenge of accurately modeling, predicting, and controlling nonlinear dynamical systems has driven the development of data-driven frameworks that blend machine learning with classical dynamical systems theory. While linear models are foundational in systems analysis and control, they fail to capture essential nonlinear phenomena such as coexisting equilibria, limit cycles, and bifurcations. In this work, we focus on bifurcations and show how classical scenarios—including saddle-node, transcritical, and pitchfork bifurcations—can be represented using linear systems. We begin by analytically demonstrating how these bifurcations arise in linear dynamical systems when viewed through a nonlinear change of coordinates, thereby constructing globally valid linear representations of classical bifurcation behavior. In this setting, bifurcations correspond to qualitative changes in the linear dynamics under a fixed coordinate transformation. Building on these insights, we show how such linear representations can be learned from data and leveraged to linearize bifurcations observed in data. The resulting framework enables data-driven identification and modeling of bifurcation behavior using linear latent dynamics, providing a new pathway for analyzing, predicting, and controlling nonlinear systems undergoing qualitative regime changes.