Linear dependence of linear transformations and images (Q910454)

Let \(A_ 0,A_ 1,...,A_ n\) be \(m'\times m\) matrices such that, for each column vector \(\alpha\) of length m, \(A_ 0\alpha\) is linearly dependent on the vectors \(A_ i\alpha\) \((i=1,...,n)\). The problem which the author studies is: under what further conditions can we conclude that the matrix \(A_ 0\) is linearly dependent on the matrices \(A_ i\) \((i=1,...,n)?\) Let \(\lambda_ 1,...,\lambda_ s\) be a basis for the column space of the matrices \(A_ 1,...,A_ n\). Then the hypothesis on the \(A_ i\) implies that there are unique row vectors \(\phi_{ij}\) of length m such that \(A_ j=\sum \lambda_ i\phi_{ij}\) for \(j=0,1,...,n\), and row vectors \(\kappa_ i\) of length n such that \(\phi_{i0}=\kappa_ iF_ i\) for \(i=1,....,s\) where \(F_ i\) is the matrix with rows \(\phi_{ij}\) \((j=1,...,n)\). The author gives a series of criteria involving the \(F_ i\) and \(\kappa_ i\) which determine when the original matrices are linearly dependent (unfortunately these are too complicated to state here). Using these criteria, he completely analyzes the situation in the \(2\times 2\) case. \(\{\) The reader should be warned that some ambiguities arise because the \(\kappa_ i\) are not uniquely determined. In particular, the author informs me that the hypotheses of Theorem 2 would be clarified by adding ``for some choice of \(\kappa_ i''.\}\)