Linear Transformation vs. Change of Basis (Deep Dive)
Last Updated on December 9, 2025 by Editorial Team
Author(s): Irene Markelic, PhD
Originally published on Towards AI.
Mastering the Math Behind AI, Machine Learning and Data Science
Linear transformations and Change of Basis are deceptively simple concepts that tempt us to rush over them quickly. However, mastering the duality between them is absolutely fundamental for understanding advanced topics like Diagonalization, SVD, and PCA. Ignoring this difference now will cost valuable time later. This article, also available here clarifies this critical duality once and for all, providing a concise conceptual overview.
The article delves into the concepts of linear transformations and changes of basis in linear algebra, emphasizing their foundational roles in advanced mathematical topics. It clearly distinguishes between how a linear transformation actively manipulates a vector within a space versus how a change of basis alters the vector’s representation without changing its position. The material is concise yet thorough, providing essential definitions, visual aids, and comparisons to clarify these key mathematical ideas, ultimately guiding the reader towards an understanding that prepares them for more complex subjects such as matrix diagonalization.
Read the full blog for free on Medium.
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