Extending Desktop over a Local Network

1.Open System Preferences from your black apple menu. Click Sharing.

2.At the top of the Services list is Screen Sharing. Select it and turn it On.

3.Where it says Allow access for:, choose All users.. This will make things easier.

4. If you are not using ScreenSharing or JollysFastVNC you have to:
Click the Computer Settings button.

From the next window check on Anyone may request permission to control screen.

On the same window enable the VNC viewres may control screen with password option and choose a password.

This allows computers running VNC software that do not support all security methods, such as Chicken of the VNC.


On your Linux machine - open your VNC client - and type in the DNS name or IP address of your mac and enter the above password

 

I think the OP wants to use the Linux box as an extra monitor a la Screen Recycler (monitor extension), not simply share (mirror) the Mac screen.

Why didn't Screen Recycler work?