Yes. Oh yes. You sure can! Boy howdy, you can do it! Yup. Uh huh. Sure.
You can indeed connect your Mac to the existing Linksys-based wireless network using an AirPort Extreme card. My home wireless network is based on a Linksys WAP54G (the access point plugs into a Linksys BEFSR41 router) and we NEVER have any problems with running around the place with our iBook. The card will indeed figure things out for itself and offer to connect to the wireless network it detects. You will (I hope) have to provide a password for the wireless network, which should not be an issue, since WPA, the new standard for WiFi security uses (gasp!) the SAME algorithm for passwords on every platform!