Suppose I have a matrix
$$
\begin{pmatrix}
1 & 1 & 0\\
0 & 1 & 1\\
0 & 0 & 1
\end{pmatrix}
$$
corresponding to a transformation $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$. How do I find the dimension of the $f$-stable subspace generated by some vector $v$?
where
$$
v=\begin{pmatrix}
1\\
2\\
3
\end{pmatrix}
$$
Dimension of f-stable subspace generated by a vector.
linear algebramatricesvector-spaces
Best Answer
You compute $f(v)$. Are $v$ and $f(v)$ linearly dependent? Then the stable $f$-subspace generated by $v$ is $1$-dimensional. Otherwise, it is at least $2$-dimensional. In order to determine whether its dimension is $2$ or $3$, start all over again: compute $f\bigl(f(v)\bigr)$. Are $v$, $f(v)$ and $f\bigl(f(v)\bigr)$ linearly dependent? If they are, then the answer is $2$; otherwise, the answer is $3$.