TITLE: Is basis change ever useful in practical linear algebra?
QUESTION [34 upvotes]: In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis?

REPLY [5 votes]: Structural dynamics with the Finite Element Method
Computing the dynamic response of a structure is done with modal analysis.
The original vector space is displacements and rotations in each node. These are unsuited to dynamic response. The system of equations is therefore transformed into a new vector space where each element in the vector space represents a vibration mode.
Wikipedia has some nice animations of vibration modes, ordered from lowest to highest frequency:






Each mode (eigenvector) is the vibration to a specific frequency (eigenvalue). When these have been found, we can compute interesting stuff like the structural response to an earthquake of a certain frequency spectrum. Afterwards, the basis is changed back to the original to obtain forces and displacements in each node.