I'm biased, but very fond of the open-access introductory textbooks used where I studied. The department was very much pure maths, but the intro classes were accessible to general liberal arts students. I think the texts are relatively unique in that they're very proof oriented, yet with a pedagogical style that doesn't assume the reader is a future graduate student.

For calculus, three options to see if you like a particular author's style of explanation: http://people.reed.edu/~mayer/math111.html/math111.pdf - the most pedagogical http://people.reed.edu/~mayer/math111.html/math111.pdf - the most beautiful https://people.reed.edu/~jerry/111/calc.pdf - the most technical

For introduction to mathematical analysis and proof: http://people.reed.edu/~mayer/math112.html/math112.pdf

For multivariable calculus: https://www.stat.rice.edu/~dobelman/notes_papers/math/calcul... [1]

For linear algebra, we used the relatively standard Friedberg Insel and Spence [2], but I hear good things about https://hefferon.net/linearalgebra/.

[1] Link is off university domain, since apparently it was at some point turned into a bit more hardcore textbook oriented towards those going onto graduate studies in mathematics. If curious: https://www.amazon.com/Calculus-Analysis-Euclidean-Undergrad...

[2] https://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/...

The Friedberg, Insel, Spence book [1] passes your test. Linear transformations first, then matrices as a representation of transformations with respect to a specified basis. Depending on your tastes, you may find it too dry, but imo it is a clean presentation.

[1]: https://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/...