If you're dealing with linked servers that have specific users specified with password then I don't think you'll be able to preserve that password information. Since linked servers are at the server level that information wouldn't be in any db backup. But, you could run queries like the one below to get the list of linked servers and check which ones use pass-through credentials. For the ones that do, just create new linked server definitions in the new database (you can easily get this SQL by right-clicking on the linked server in SSMS and generating the script). For the ones that don't, you can create the linked server via TSQL but you'll have to give them new credentials.
SELECT
serv.NAME,
serv.product,
serv.provider,
serv.data_source,
serv.catalog,
prin.name,
ls_logins.uses_self_credential,
ls_logins.remote_name
FROM
sys.servers AS serv
LEFT JOIN sys.linked_logins AS ls_logins
ON serv.server_id = ls_logins.server_id
LEFT JOIN sys.server_principals AS prin
ON ls_logins.local_principal_id = prin.principal_id