In my experience, Linear Algebra Done Right by Axler is a great second book on linear algebra. But it makes sense to most students only if they've seen and done concrete matrix calculations before. Giving it to a mathematical novice is likely to lead to baffelment. Too abstract.
I don't think it's too abstract. It contains plenty of geometrical intuition. And it makes things very clear and he says them very _early_. E.g., a matrix represents a linear transformation. It took me ages to understand this at high school because I was lost in a sea of matrices.
