To a student in introductory physics, using vectors is at best an exercise in bookkeeping. A two-dimensional kinematics problem effectively doubles the number of equations that a student must know, and invites the student to memorize factoids: "The horizontal motion is constant," "Gravity is only in the y-direction," etc. Force problems prompt the drawing of vectors on free-body diagrams, but are solved algebraically using components only. This focus on vectors from a components-only standpoint can lead students to develop a poor intuitive understanding of the differences between vectors and scalars, and it has been my experience that a poor understanding of vectors carries over into upper-level coursework by physics majors as well. In recent years, I have developed a pedagogical method in my own classes that may help students visualize and understand vectors as different from scalars using linear algebra notation. In this paper, I will describe this method using two-dimensional vectors as appropriate for an algebra-based general physics course.