This study contributes to the literature on linear algebra instruction by designing and researching a teaching sequence based on APOS Theory to introduce engineering students to vector spaces. The sequence offers students multiple opportunities to understand the concept. Another contribution is the evidence that introducing prerequisite concepts--such as equality, sets, and binary operations--before tackling vector space was crucial for grasping the role of proof in determining whether a set is a vector space. The findings confirm that, as other studies have shown, vector space is challenging for students. However, the results demonstrate that students were able to face this challenge. All students showed evidence of developing an understanding of the concept, with nine achieving a clear grasp of vector space and the role of proof by the end of the experience. Additionally, the progress observed--from having difficulties with symbols to successfully proving statements involving unconventional operations--underscores the effectiveness of the teaching approach.