Many of the central ideas in an introductory undergraduate linear algebra course are closely tied to a set of interpretations of the matrix equation Ax = b (A is a matrix, x and b are vectors): linear combination interpretations, systems interpretations, and transformation interpretations. We consider graphic and symbolic representations for each, highlighting the very different ways that the vector x is conceived across interpretations. We illustrate how a framework consisting of these interpretations can be used to make sense of student thinking using data from mid-semester interviews with introductory linear algebra students at a large public university in the southwestern US.