This paper investigates the stability and stabilization of continuous-time cross-dimensional linear systems (CCDLSs) with dimension-bounded. Firstly, by V-product and V-addition of matrices and vectors, a new approach is proposed for calculating the solution of CCDLSs. Secondly, using this solution, one necessary and sufficient condition is established for determining whether a CCDLS is asymptotically stable with a given initial state, based on which, all initial states that make the CCDLS asymptotically stable can be found. Thirdly, all initial states that can be stabilized are discussed and an algorithm for designing the corresponding controller is given. At last, two examples are given to show the feasibility of the theoretical results.