This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we provide some details and proofs of some results from (Strang, 1993). We then describe the fundamental theorem of linear algebra from different views and find the properties and relationships behind the views. The fundamental theorem of linear algebra is essential in many fields, such as electrical engineering, computer science, machine learning, and deep learning. This survey is primarily a summary of purpose, significance of important theories behind it. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in theory behind the fundamental theorem of linear algebra and rigorous analysis in order to seamlessly introduce its properties in four subspaces in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results and given the paucity of scope to present this discussion, e.g., the separated analysis of the (orthogonal) projection matrices. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields. Some excellent examples include (Rose, 1982; Strang, 2009; Trefethen and Bau III, 1997; Strang, 2019, 2021).
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