What are eigenvalues associated with in linear transformations?

Eigenvalues are closely associated with linear transformations in that they represent how a vector is scaled when that linear transformation is applied. Specifically, when a linear transformation is described by a matrix, eigenvalues quantify the factor by which an eigenvector is stretched or compressed during the transformation.

If you consider the linear transformation represented by a matrix ( A ), an eigenvalue is a scalar ( \lambda ) such that when ( A ) acts on an eigenvector ( v ), the result is equivalent to scaling that eigenvector by ( \lambda ). This can be expressed mathematically as ( A v = \lambda v ). Thus, the eigenvalue gives you an idea of how the eigenvector is affected in terms of scale.

In contrast, the other choices touch on different concepts in linear algebra. The number of solutions to an equation is related to system consistency and rank, vector dimensions pertain to the concept of vector spaces, and calculating matrix determinants is a procedure used to glean properties of matrices but does not directly relate to the meaning of eigenvalues.