What happens to eigenvectors under linear transformations?

Eigenvectors are special vectors associated with a linear transformation represented by a matrix. When a linear transformation is applied to an eigenvector, the result is a new vector that is a scaled version of the original eigenvector. The scaling factor is the corresponding eigenvalue associated with that eigenvector. This means that the direction of the eigenvector remains unchanged, while its magnitude may increase or decrease depending on the eigenvalue.

For example, if ( A ) is a matrix and ( v ) is an eigenvector of ( A ) with eigenvalue ( \lambda ), the relationship is expressed as:

( A v = \lambda v )

This indicates that transforming ( v ) by the matrix ( A ) merely scales ( v ) by ( \lambda ), rather than changing its direction. If ( \lambda ) is positive, the eigenvector retains its direction, while if ( \lambda ) is negative, the direction is reversed but remains collinear. Therefore, it is accurate to say that under linear transformations, eigenvectors are scaled by their eigenvalues, which is the reason for selecting this option as correct.