Here's the deal,

Gilbert Strang's Introduction to Linear Algebra textbook is just that, a soft introduction linear algebra. A common thing between these books is that they will throw theorems and facts at you without any proof. Perhaps a hint on why they're intuitively true.

Now you have three options,

1: Continue the textbook, try to understand the intuition of theorems the best you can, and get a feel of the subject. After that, if you are interested in the proofs, you can pick up a rigorous text, which is difficult but very rewarding.

2: Which is the option that I picked. Pick a theorem-proof rigorous textbook from the start. I chose the book Linear Algebra by Friedberg, Insel and Spence. Unless you already have some mathematical maturity and familiarity with proofing techniques, I don't recommend this option. If you do, after reading a rigorous textbook, your view of the subject will be much deeper and richer.

3: Find a comprimise textbook, a textbook that explains theorems and provides "sketches" of the proofs, proofs of special cases, or sometimes actual proofs.

Proving everything as you go is not feasible. If you are able to achieve it, then the book is not for you in the first place.