By introducing matrix multiplier and vector multiplier two kinds of semi‐tensor products (STPs), called matrix–matrix (MM) STPs and matrix–vector (MV) STPs, are introduced. They are generalisations of conventional matrix product, and contain standard STP as a particular case. Certain properties are revealed. Using consistent MM‐STP and MV‐STP, a cross‐dimensional linear system is proposed. It is shown that the cross‐dimensional linear system is a linear semi‐group (S) system. Next, the quotient matrix space based on matrix multiplier and the quotient vector space based on vector multiplier are introduced, and the S‐system structure is extended to quotient matrix and quotient vector spaces. Finally, an inner‐product is introduced, which poses a topology on the cross‐dimensional state space and hence turns the linear S‐system into a dynamic linear system.