Analysis is Calculus, presented formally. The idea of Analysis is that you may have learned calculus algorithms already in high school (or college), but you might not have learned formal math proofs, so your first "real math" class is a re-do of calculus where you focus on carefully proving all those things you already learned the mechanics of.

If you've learned Calculus, and you've learned introduction to proofs (either on your own, or in summer nerd camp, or by visiting a local college), you are ready to learn new material (like linear algebra) in a proof-focused curriculum.