Interpretation of market completeness: full row rank payoff matrix

Question

Suppose that there are $K$ assets and $S$ states of nature. The assets' payoff is represented by the matrix $$ \underbrace{R}_{S\times K}=\begin{pmatrix} r_{11}&\cdots& r_{K1}\\ \vdots&\ddots&\vdots\\ r_{1S}&\cdots& r_{KS} \end{pmatrix}, $$ that is the $k$-th column of $R$ is the payoff vector $r_k$ of asset $k$. If $z$ is a portfolio of assets ($z$ is dimension $K\times 1$), then the payoff of $z$ is $Rz$.

The market is complete if $R$ has rank $S$ (i.e. full row rank). Is there an intuition for the word 'complete' in this context? I think I used to understand this using span but now I am getting confused. Can someone help me please?

Answer 1

It means that it is possible to construct a complete set of state contingent claims. For any $x$ ($x$ is dimension $S\times1$) there exist a $z$ such that $x=Rz$.

Answer 2

Market is complete means, by definition, that $R$ is surjective, i.e. has full row rank. In other words, given any payoff $(x_1, \cdots, x_S)$, there exists a portfolio $z$ such that

$$ x = Rz. $$

Now $R$ is surjective iff $R^T$ is injective, i.e. has full column rank. So equivalently, market completeness means that the payoff vectors, indexed by states, are linearly independent. In particular, if there are more states than assets, market cannot be complete.