The span of a set of vectors is the set of all possible linear combinations of those vectors. It is denoted by span(v1, v2, …, vn) or Span(V), where V is the set of vectors {v1, v2, …, vn}.
Formally:
Given a set of vectors V = {v1, v2, …, vn} in a vector space, the span of V is defined as:
span(V) = {v | v = a1v1 + a2v2 + … + anvn, where ai ∈ ℜ}
In other words, the span of V consists of all vectors that can be expressed as a linear combination of the vectors in V, where the coefficients ai are real numbers.
Key properties:
• The span of a set of vectors is a subspace of the original vector space.
• The span of a set of vectors contains all the original vectors.
• The span of a set of vectors is the smallest subspace that contains all the original vectors.
Geometric interpretation:
- In 2-dimensional space, the span of two non-parallel vectors is the entire plane.
- In 2-dimensional space, the span of two parallel vectors is a line.
- In 3-dimensional space, the span of three non-coplanar vectors is the entire space.
- In 3-dimensional space, the span of two non-parallel vectors is a plane.
- In 3-dimensional space, the span of two parallel vectors is a line.
Examples:
- The span of the standard basis vectors {i, j} in 2-dimensional space is the entire plane.
- The span of the standard basis vectors {i, j, k} in 3-dimensional space is the entire space.
- The span of the vectors {v1, v2} is the line or plane that contains both v1 and v2, depending on whether they are parallel or not.
This definition and explanation of span can be a useful addition to your Zettlekasten notes on linear algebra. You can connect it to other concepts, such as linear independence, basis, and dimension, to deepen your understanding of the subject.
Shear