Coordinates
A vector is represented by the column .
How is represented?? Change of basis? (Open questions in the notes.)
Direct sums
if the span (for every there exist such that ) and are independent. Then
is an isomorphism.
and , i.e.
Theorem N21.1 (Nullity–rank).
Duality
Definition N21.2 (Dual space). The dual of is .
Definition N21.3 (Dual basis). Let be a finite ordered basis of . Then the dual basis of consists of the such that
When they are “orthogonal”, so “sees” no ; when they are aligned together.
The map
is an isomorphism if (finite-dimensional ), and injective if .
Definition N21.4 (Annihilator). For , the annihilator of is
Example N21.5. For on , is the -coordinate of : .
With the identity matrix, in suitable bases
where the top block of rows corresponds to and the right block of columns to .
Definition N21.6 (Transpose). The transpose of is , .
If :
and
Quotients, invariant subspaces and eigenvectors
Quotient: cosets.
Definition N21.7 (Invariant subspace). For , a subspace is invariant if , i.e. .
Definition N21.8 (Eigenspace). The eigenspaces of are .
Consider a basis of eigenvectors , with , so . Let be ; then is some scalar times , i.e. for some .
The generalized kernel is for all .
Let’s find all eigenvectors for ! Then
so is an eigenvector for .
Nilpotent maps
Definition N21.9 (Nilpotent). is nilpotent if for some .
Example N21.10. On , let , with matrix . Then
So , and the index of nilpotency is .
Orbit counting