You might want to try Linear Algebra Done Right by Sheldon Axler. It's a short book, succinct but extremely clear and approachable. It explains Linear Algebra without using determinants, which are relegated to the end, and emphasises understanding the powerful ideas underpinning the subject rather than learning seemingly arbitrary manipulations of lists and tables of numbers.
Those manipulations are of course extremely useful and worth learning, but the reasons why, and where they come from, will be a lot clearer after reading Axler.
As someone pointed out elsewhere in this thread, the book is available free at https://linear.axler.net/
The page count suggests that we have different ideas of what's meant by "short". In any case, it looks great from the forewords. If I ever want to make a serious try to really get it, this is probably what I'll use.
It is widely considered to deliver on the promise of done right!