  Dimension-varying linear systems are investigated. First, a dimension-free
state space is proposed. A cross dimensional distance is constructed to glue
vectors of different dimensions together to form a cross-dimensional
topological space. This distance leads to projections over different
dimensional Euclidean spaces and the corresponding linear systems on them,
which provide a connection among linear systems with different dimensions.
Based on these projections, an equivalence of vectors and an equivalence of
matrices over different dimensions are proposed. It follows that the dynamics
on quotient space is obtained, which provides a proper model for
cross-dimensional systems. Finally, using lifts of dynamic systems on quotient
space to Euclidean spaces of different dimensions, a cross-dimensional model is
proposed to deal with the dynamics of dimension-varying process of linear
systems. On the cross-dimensional model a control is designed to realize the
transfer between models on Euclidean spaces of different dimensions.
