2. Generate a space from one seed

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The same vector space can require different generating data under different allowed operations. Here the allowed operations are real linear combinations of powers of one matrix, including the identity.

Choose the action and a seed

On \(\mathbb R^2\), let

\[N=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad v=\begin{pmatrix}0\\1\end{pmatrix}. \]

Apply the operator repeatedly:

\[N^0v=v=\begin{pmatrix}0\\1\end{pmatrix},\qquad Nv=\begin{pmatrix}1\\0\end{pmatrix},\qquad N^2v=\begin{pmatrix}0\\0\end{pmatrix}. \]

In fact \(N^2=0\), so further applications give zero too.

Recover every vector

For any real numbers \(a,b\),

\[\begin{pmatrix}a\\b\end{pmatrix} =b\,v+a\,Nv=(bI_2+aN)v. \]

Thus one seed generates the whole space under this action. For the input \((2,3)^{\mathsf T}\), the coefficients are \(b=3\) and \(a=2\).

An empty seed set spans only the zero vector. Our space is nonzero, so one is also the minimum number of seeds for this particular action.

A poor seed and a different action

If we choose \(w=(1,0)^{\mathsf T}\) instead, then \(Nw=0\). Its generated space contains only multiples of \(w\), so it misses \((0,1)^{\mathsf T}\).

Now change the allowed operator to the zero matrix. Every positive power of that operator kills every vector. The identity still preserves the supplied seeds, but no new directions appear. One seed spans at most a line, while the two standard basis vectors span \(\mathbb R^2\). The minimum is now two.

The seed count therefore belongs to the space and its declared action. It is not automatically the dimension of the vector space.

This is the classical cyclic-vector viewpoint: a linear operator defines a polynomial-module action. See Paul Garrett's Abstract Algebra (June 2007), §10.4, especially Corollary 10.4.1 and Example 10.4.3. The sources and contributions guide places this tutorial within Paper I's broader development.

Try it

Which single seeds \(v=(a,b)^{\mathsf T}\) generate \(\mathbb R^2\) under \(N\)?

Answer: Exactly those with \(b\ne0\). Here \(Nv=(b,0)^{\mathsf T}\), and the matrix with columns \(v,Nv\) has determinant \(-b^2\). It is invertible precisely when \(b\ne0\). If \(b=0\), both vectors lie on the first coordinate axis.

Continue reading: Span, seed rank, and the seed operations guide. Paper I's broader definitions may use modules, topology and cardinalities.

Papers, proofs, Lean, and code

Terms in this example: Seed · Seed rank · Cyclic vector · Span · Operator algebra

Paper I develops seed generation with its stated algebra and module hypotheses.

Paper I: Typed Operator Systems, Seed Rank, and Exact Packet Coordinates

Read · Written proofs · Lean coverage · Code and setup · Evidence

Related library guide and code examples. The guide covers related constructions and their software interfaces.

Python arithmetic check for this example: seeds(). This checks the displayed rational, modular or finite-set calculations exactly.

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