september 4

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)

Linear operators

Their range=image, null-space=kernel (both linear subspaces).

Rank=dimension of the image (if finite-dimensional).

Important: dimension of the kernel + rank=dimension of the domain…

Example:

\(V=\Fun(S,\Field)\), where, as before \(S=\{0,1,2,3,4,5\}\subset\Comp\). Let \(Z\) is the operator of multiplication by \(z\). What is the kernel? Image? What about differentiation?

Matrices

A linear operator
\[
A:V\to W,
\]
and bases \(\f,\e\) in \(V,W\) yield the matrix of linear operator \(A\):if
\[
A(f_l)=\sum_l A(k,l)e_k,
\]
then the matrix is
\[A=
\left(
\begin{array}{ccc}
A(1,1)&\ldots &A(1,n)\\
\vdots &\ddots &\vdots \\
A(m,1)& \ldots&A(m,n)\\
\end{array}
\right).
\]

What happens if one changes the bases in \(V\) and \(W\), replacing \(\e’\mapsto \e\) and \(\f\mapsto \f’\)? Nothing unexpected.

What happens if \(V=W\), \(\e’=\f’\)?
Similarity transformation: if in the basis \(e\) of \(V\) the matrix of a linear operator is \(A\), and \(P\) is the matrix of basis change from \(\f\) to \(\e\) (i.e. \(\e_k=\sum_lP(k,l)\f_l\)), then the matrix of the same transformation in the basis \(\f\) is
\[
P^{-1}AP: (AP)_{kn}=\sum_l (P^{-1})(k,l)A(l,m)P(m,n).
\]

Similarity transformations do not change the operator, but its matrix. Matrix can be made simpler, in some sense.

Example

multiplication by \(z\) on \(S\subset\Comp\) as above.
Hint:
\[
z(z-1)(z-2)(z-3)(z-4)(z-5)=-120 z + 274 z^2 – 225 z^3 + 85 z^4 – 15 z^5 + z^6.
\]

Example

Laplace operator: \(n\) altruists stand in a circle, and at each step each of them splits what she has in two and gives to her neighbors, left and right.

Eigenvectors (of a linear operator \(A\)) are the elements, on which the action of the operator is especially easy:
\[
Av=\lambda v.
\]

Eigenvalues: roots of the characteristic polynomial
\[
\det(A-\lambda E).
\]
If all roots are simple (multiplicity one), then the linear operator is isomorphic to multiplication by \(z\) on \(\Fun(\sigma(A))\).

If not – Jordan normal form

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