<p>This paper investigates the problems of partial state consensus and output consensus for heterogeneous linear multi-agent systems (MASs). Firstly, the partial state consensus problem of parameter heterogeneous linear MASs is solved by converting it to a corresponding partial stability problem via the linear transformation approach, a necessary and sufficient condition for achieving partial state consensus is obtained utilizing the partial variable stability theory and a bilinear matrix inequality (BMI)-based algorithm for finding the gain matrices in the control protocols is presented. Secondly, the partial state consensus problem of structural heterogeneous linear MASs with distinct state dimensions is dealt with, and a necessary and sufficient condition is derived by a similar technical route. Finally, the output consensus problem of heterogeneous linear MASs is considered and a necessary and sufficient condition is derived by a linear transformation to convert the output consensus problem to the partial state consensus problem. The obtained results are verified through several numerical examples.</p>