Wow! Your textbook looks phenomenal. Eyeballing the table of contents and a few chapters, this arrangement of material looks exactly like what I'm advocating throughout this thread. Gaussian elimination and systems of linear equations before vector spaces, a coverage of fields in tandem with vector spaces, some treatment of sets and functions in appendices, generous usage of matrices to supplement theory...

And you include solutions for your exercises? This is fantastic! Do you have any plans to write something like this for functional analysis?