Determining the Span of Three Vectors in R3

Given three vectors from $\mathbb{R}^3$, which are (2, 1, -1), (0, 2, 2), and (-1, -1, -1), determine their span by forming the linear combination $a\mathbf{v_1} + b\mathbf{v_2} + c\mathbf{v_3}$ where $a, b,$ and $c$ are scalars.

In this problem, we're exploring the concept of 'span' in linear algebra. The span of a set of vectors is the collection of all possible linear combinations of those vectors. In the context of this problem, we have three vectors in three-dimensional space, and we are tasked with determining what subset of the space is described by these vectors. This involves considering all possible combinations of the vectors using different scalars. Conceptually, spanning vectors can be thought of as providing a 'basis' or foundation upon which the rest of the space is built, though this specific set of vectors may not be a basis in the formal sense unless they are linearly independent.

Understanding vector span is a critical step toward exploring other fundamental concepts in linear algebra like linear independence, basis, and dimension. The span quantitatively describes what portions of a vector space are covered by given vectors and helps identify if a vector set is capable of covering an entire space or just a subset. If the vectors span the full space, they are said to form a basis for that space, provided they are also linearly independent.

Additionally, this problem serves as a prerequisite to grasping more complex topics like linear transformations and matrix operations. By mastering the concept of the span, you gain insights into how complex vector spaces can be manipulated, transformed, and understood in higher mathematical contexts.