You are trying to connect an AP to a wireless router. That does not work. Incompatible plumbing.
A wireles router IS an AP. An AP with a router, switch, firewall and DHCP server added to it.
A wireless router creates a wireless network. An AP creates a wireless network. CLIENTS associate with APs and wireless routers.
What you want is a bridge. A bridge is a client that may have a router, switch, firewall and DHCP server added to it.
If your Linksys is compatible with Sveasoft, DD-WRT, Open WRT or other third party firmware, you can make that happen with a firmware upgrade.