Change of basis

Change of basis

Change of basis refers to the process of transforming the coordinates of a vector from one basis to another. This transformation is often represented by a matrix, and in a linear algebraic setting, it is equivalent to a coordinate transformation. The way in which the components of a vector change under a change of basis can be determined through this process.From: Applied Structural and Mechanical Vibrations [2019], The negative exponential transformation: a linear algebraic approach to the Laplace transform [2023]

Orthogonality

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Published in Crista Arangala, Exploring Linear Algebra, 2019

Crista Arangala

For example, for ui in B2, ui = k1v1 + k2v2 +…+ knvn where k1,k2,…, kn are scalars. And thus you will see in changing from basis B1 to basis B2 there are n equations. The matrix of scalar constants resulting from this change of basis is called the change of coordinates matrix.

Vector Spaces

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Published in Lina Oliveira, Linear Algebra, 2022

Lina Oliveira

be the change of basis matrix from the standard basis P1 to basis the B.

Linear Algebra

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Published in Niket S. Kaisare, Computational Techniques for Process Simulation and Analysis Using MATLAB®, 2017

Niket S. Kaisare

In systems where the matrix multiplication by A maps the vector space on itself, change of basis is represented by similarity transform. This was seen in Example 2.5. Change of basis therefore does not change the eigenvalues of the system.

The negative exponential transformation: a linear algebraic approach to the Laplace transform

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Published in International Journal of Mathematical Education in Science and Technology, 2023

Wha-Suck Lee

The concept of an exponential transformation is deeply rooted in the idea of a coordinate transformation where the original problem is translated into a different space. In a linear algebraic setting, coordinate transformations take the form of a change of basis. The translation of a power series into a Laplace transform via a negative exponential transformation opens up a linear algebraic approach to the Laplace transform. The importance of a negative exponential transformation for the Laplace transform is validated by the Taylor inversion of the Laplace transform.

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Orthogonality

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The negative exponential transformation: a linear algebraic approach to the Laplace transform