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60 pages before defining vector spaces seems like a not so good approach to linear algebra. Since Linear Algebra Done Right has already been recommended, I will suggest Linear Algebra Done Wrong: http://www.math.brown.edu/~treil/papers/LADW/LADW.html

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I know, I would have liked to have linear transformations and vector spaces moved forward, but without the computational skills (grunt work) you can't really do much with vector spaces and lin. trans. except define them.

Hence the current ordering of chapters...

Then just define them! The entire subject is about linear transformations on vector spaces. The biggest problem with linear algebra as its taught today is that students uniformly come out with an understanding that linear algebra is about matrices. It should be made clear that a matrix is simply a representation of a transformation, and then when you do teach the grunt work operations, you can teach them as what they actually are (multiplication: composition of transformations, determinant: area of unit square under transformation, etc.)

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