I write as someone who has taught calculus/linear algebra to first year engineering/science students in the US and Spain. In a typical European system (and probably most of the rest of the world) a student studying a technical area (science, engineering, medicine) in the university will have seen in the equivalent of high school a full year of calculus and a full year of some sort of matrix calculus course that involves using Gaussian elimination to solve systems of linear equations. In the US even a student studying a technical area may have seen in high school neither calculus nor matrices. Probably more students will have seen calculus than not, but very few will have seen a systematic treatment (meaning using elimination, involving formalizing parameter counts in terms of rank and dimension, etc.) of the solution of systems of linear equations.
For example, in Spain one can reasonably assume that first-year engineering students know that the solutions of a system of $m$ linear equations in $n$ variables depend on $n-r$ parameters, where $r$ is the rank of the system, and, moreover, that they can compute a parametric description of these solutions via row reduction. This material appears on the exams used to determine placement in the university, and is covered in the last year of high school. One does have to review this material in a first semester linear algebra class, because they probably do not know it as well as they should, nor have they seen it presented with much sophistication (their understanding is purely operational). In the US, teaching engineering students (the situation might be different teaching math majors) one cannot suppose familiarity even with the matricial representation of a system of linear equations, nor even that students have seen systems of linear equations involving more than two variables (which they were taught to solve "by hand"). Also linear algebra would usually be a second or third semester course, not a first semester course.
The conservative assumption is that US students entering the university have seen no linear algebra (to the point of not even knowing what a matrix is) and have not learned much more than very mechanical manipulation of derivatives and integrals. With non-technical students the conservative assumption is that they do not know basic trigonometry and with polynomials can do little more than factor two variable polynomials. It is also important to remember that their backgrounds are far more heterogeneous than they would be in many other systems. The US "system" is not a system at all, and it is a mistake to assume uniformity of preparation. Moreover, even for nominally well prepared students the expected level is less than what it is in many other countries. On the other hand, the flexibility underlying this heterogeneity also means that an occasional talented student will have studied outside the standard curriculum, and will have learned linear algebra, vector calculus and more, but often that student gets channeled into level appropriate courses in the university too.