with the hdmi cable disconnected can you run the command
dmesg -w
in a terminal. that command should allow you to see if your computer is recognizing the tv when the cable is plugged in. after running the command if you would connect the hdmi cable to the tv and let us know if the output from dmesg changed to show the connection.
if the tv is recognized and you get more messages from dmesg, it might help to copy and paste that output into your reply.