Definition of eigenvector

Question

Eigenvector is such a vector which gets stretched (elongated or compressed by scaling factor of eigenvalue) when linear transformation A is applied to it. Please correct me if wrong

I am trying to understand it geometrically. given a linear map f or A, for that map, in V vector space, i will search for vectors such that those vectors gets streched / scaled if i map that vector using the given linear transformation. Such vectors in V which gets scaled after f - mapping, those vectors are called eigen vectors. Pls tell whether i am thinking correctly or not.

Answer 1

It is a bit unclear of what you are asking, but here is an attempt to remove some of your doubts.

Definition - Eigenvalue and Eigenvector

Let $f: V \rightarrow V$ be a linear map of the vector space $V$ into itself. If a proper nonzero vector $\mathbf{v}\in V$ and a scalar $\lambda$ exist such that $$f(\mathbf{v})=\lambda \mathbf{v}, $$ then the proportionality factor $\lambda$ is called an eigenvalue for $f$, while $\mathbf{v}$ is called an eigenvector belonging to $\lambda$.