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Stapler2000
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Nevermind -- Polygons and Polywags.
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A linear transformation is a mathematical function that maps a vector space to itself, preserving its linear structure. In simpler terms, it is a transformation that does not change the shape of an object, only its size and orientation.
Diagonalization is a technique used in linear algebra to simplify and better understand a linear transformation. It involves finding a new basis for the vector space in which the transformation has a diagonal matrix representation, making it easier to perform calculations and analyze the properties of the transformation.
Diagonalization is closely related to eigenvalues and eigenvectors, as the process of diagonalization involves finding a basis of eigenvectors for the given linear transformation. The eigenvalues are then used to construct the diagonal matrix representation of the transformation.
A linear transformation is diagonalizable if it can be represented by a diagonal matrix, while a non-diagonalizable transformation cannot be represented in this form. In other words, a diagonalizable transformation has a full set of linearly independent eigenvectors, while a non-diagonalizable transformation may have repeated eigenvalues or a lack of eigenvectors.
Diagonalization has many applications in various fields such as physics, engineering, and economics. It is used to simplify and better understand complex systems, such as circuits, differential equations, and financial models. It also plays a crucial role in solving optimization problems and in data analysis and machine learning techniques.