Transformer training is notoriously difficult, requiring a careful design of optimizers and use of various heuristics. We make progress towards understanding the subtleties of training Transformers by carefully studying a simple yet canonical linearized shallow Transformer model. Specifically, we train linear Transformers to solve regression tasks, inspired by J.~von Oswald et al.~(ICML 2023), and K.~Ahn et al.~(NeurIPS 2023). Most importantly, we observe that our proposed linearized models can reproduce several prominent aspects of Transformer training dynamics. Consequently, the results obtained in this paper suggest that a simple linearized Transformer model could actually be a valuable, realistic abstraction for understanding Transformer optimization.