Vector resolution is the process of graphically or trigonometrically determining the magnitude and direction of a vector's components. Vectors - Motion and Forces in Two Dimensions - Lesson 1 - Vectors: Fundamentals and Operations. Vector Resolution.

The columns - or rows - of a rank r matrix will span an r-dimensional space. If r=3 and the vectors are in R^3, then this must be the whole space. However, that's not the only way to do it. For example, you could look at the null space, and use the rank-nullity theorem.