an illustration question in Meyer's book I am reading the "Nilpotent matrices and Jordan structure" chapter of Meyer's "Matrix analysis and applied linear algebra". I just do not quite understand the  for illustrating extending basis $S_i$ for the subsspace chain $M_i$'s. Could anyone give some explanation?
If this is a 2D vector space, the subspaces (except the trivial ones) are lines passing origin. Its basis is then only one vector along such line (for example, can be taken as the unit vector with such direction.) So what do all those parallelograms/colored dots mean here?
 A: It is not supposed to be a 2D space. It is supposed to illustrate a nested sequence of subspaces of the nullspace. All the subspaces $\mathcal M_4, \mathcal M_3, \mathcal M_2, \mathcal M_1, \mathcal M_0$ are subspaces of the nullspace of the matrix $L$ satisfying $\mathcal M_4 \subseteq \dots \subseteq M_0 = N(L)$. 4 is here is the largest exponent such that $L^k \neq 0$. Every dot in the figure is supposed to be a basis vector.
The text describes it by first finding a basis for $M^4$, then extending it to $M^3$, etc. until you reach $M^0 = N(L)$ and then have a basis for the whole nullspace.
The next picture (7.7.2) then describes how these vectors are extended to a whole basis for $\mathbb C^n$. If the vector $b$ is represented by a dot in $\mathcal M_i$ (but not $\mathcal M_{i+1}$), you can build a chain "on top" of this vector by solving $b = L^i x$ and then taking $L^{i-1}x, L^{i-2}, \dots, L x, x$ to be the chain formed by $b$. Thus, if $b$ is a basis vector for $\mathcal M_i$ (but not $\mathcal M_{i+1}$) it will have a tower of $i$ vectors on top of it (in fig 7.2.2).
