Basic question about transaction verification I am fairly new to crypto and had some questions about how transactions are verified by the nodes
Would it be correct to say that the full nodes check to see if the sender(Alice’s) signature is valid. And they check against double spends by seeing if the hash value that is included (of the transaction where Alice received the bitcoin) has ever been used an another transaction?
My question is, if that is the case, why can’t the verification only check against transactions signed by Alice, why does it have to check against the entire blockchain?
Why can’t it just check against Alice’s other transactions since you know she controls that coin.
My question is, if that is the case, why can’t the verification only check against transactions signed by Alice, why does it have to check against the entire blockchain?

A node has no concept of who controls certain outputs. So there is no way to 'check all of Alice's outputs', how would the node be able to ascertain which outputs belong to her?
Rather, the normal operation of a node is to check all transactions, to ensure that no transaction has spent coins that were previously spent (and to check that new coins aren't being created against the network's rules, etc). In doing this check, the node creates an index of unspent transaction outputs (UTXOs), which it will use to help check validity of new transactions.