[autopilot23]

I don't think reading one book is enough to learn linear algebra given its breadth and applicability. However the approach used in that one is very helpful if you want to learn functional analysis later on. But still, it probably should be read with other books that have more geometric flavour. Determinants are also a big magical topic of their own and show up in many areas of mathematics, so it would be a shame to banish them from the start like that.