The quotient map \(\pi\) is the Linear Map \(V \to V / U\) such that:

\begin{equation} \pi(v) = v+U \end{equation}

for \(v \in V\).

I.e.: the quotient map is affine subsetification map given a vector.

Suppose \(T \in \mathcal{L}(V)\), and \(U \subset V\), an invariant subspace under \(T\). Then:

\begin{equation} (T / U)(v+U) = Tv+U, \forall v \in V \end{equation}

where \(T / U \in \mathcal{L}(V / U)\)

“if you can operator on \(V\), you can operator on \(V / U\) in the same way.” Yes I just verbed operator.

quotient operator is well-defined

Why is this not possible for any subspace of \(V\)? This is because we need \(T\) to preserve the exact structure of the subspace we are quotienting out by; otherwise our affine subset maybe squished to various unexpected places. The technical way to show that this is well-defined leverages the property of two affine subsets being equal:

A quotient space is the set of all affine subsets of \(V\) parallel to some subspace \(U\). This should be reminiscent of quotient groups.

constituents

• vector space \(V\)
• a subspace \(U \subset V\)

requirements

\begin{equation} V / U = \{v+U : v \in V \} \end{equation}

additional information

operations on quotient space

Addition and scalar multiplication on the quotient space is defined in the expected way:

given \((v+U), (w+U) \in V / U\), and \(\lambda \in \mathbb{F}\):

\begin{equation} \begin{cases} (v+U) + (w+U) = ((v+w)+U) \\ \lambda (v+U) = ((\lambda v)+U) \end{cases} \end{equation}

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