Span

Definition

In linear algebra, the span of a set of vectors is the collection of all possible linear combinations of those vectors. This concept is essential as it defines a vector space generated by the vectors, showing all points that can be reached by scaling and adding them together. Understanding the span helps in determining the dimensionality of vector spaces and how they relate to each other.

5 Must Know Facts For Your Next Test

1. The span of a single vector is a line through the origin in the direction of that vector.
2. If you have two non-parallel vectors in two dimensions, their span covers the entire two-dimensional plane.
3. The dimension of the span corresponds to the number of vectors in a linearly independent set that can be used to generate that span.
4. If the set of vectors is linearly dependent, the span remains unchanged by removing some of those vectors.
5. In three dimensions, three non-coplanar vectors can generate the entire three-dimensional space when combined.

Review Questions

• How does the concept of span relate to linear combinations and vector spaces?
  • The concept of span is directly tied to linear combinations, as it includes all possible combinations formed by scaling and adding vectors. By understanding how to create these combinations, one can identify which points are reachable within a given vector space. This relationship highlights how spans define the boundaries and dimensions of vector spaces based on the vectors available.
• In what scenarios would removing a vector from a set still result in the same span?
  • Removing a vector from a set will not affect the span if that vector is linearly dependent on the others. This means that it can be expressed as a combination of the remaining vectors, so its removal does not change the overall coverage of the span. Understanding this helps simplify calculations and determine the minimum necessary vectors to maintain a given span.
• Evaluate how understanding spans can impact solving systems of equations in economics.
  • Understanding spans plays a crucial role in solving systems of equations, especially when determining feasible solutions in economic models. By recognizing which combinations of variables (vectors) create valid outcomes (spans), economists can analyze constraints and optimize resource allocation effectively. Additionally, this knowledge assists in understanding relationships between multiple economic factors, allowing for more robust decision-making based on available data.

Related terms

Linear Combination: A linear combination is an expression formed by multiplying each vector in a set by a scalar and then adding the results together.

Vector Space: A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars, satisfying certain axioms.

Basis: A basis is a set of vectors in a vector space that are linearly independent and span the entire space, meaning any vector in the space can be expressed as a linear combination of the basis vectors.