Axler 5.A

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 eigenspaces are disjoint operators on finite dimensional V has at most dim V eigenvalues quotient operator is well-defined Questions for Jana Interesting Factoids “eigenvalue” is sometimes called the “characterizing value” of a map
finding eigenvalues with actual numbers natural choordinates of a map