EIGENSTUFF and OPERATORS! Invariant subspaces are nice.

Sometimes, if we can break the domain of a linear map down to its eigenvalues, we can understand what its doing on a component-wise level.

## Key Sequence

- we defined an invariant subspace, and gave a name to 1-D invariant subspaces: the span of eigenvectors
- we showed some properties of eigenvalues and showed that a list of eigenvectors are linearly independent
- a correlate of this is that operators on finite dimensional V has at most dim V eigenvalues

- finally, we defined map restriction operator and quotient operator, and showed that they were well-defined

## New Definitions

- invariant subspace
- conditions for nontrivial invariant subspace

- eigenvalues + eigenvectors + eigenspace
- two new operators: map restriction operator and quotient operator

## Results and Their Proofs

- properties of eigenvalues
- list of eigenvectors are linearly independent
- quotient operator is well-defined

## Questions for Jana

## Interesting Factoids

“eigenvalue” is sometimes called the “characterizing value” of a map