Span

Definition

In mathematics, the span of a set of vectors is the collection of all possible linear combinations of those vectors. It represents a subspace formed by these combinations and captures the idea of reaching every point in that subspace through those vectors. The concept of span is fundamental in understanding vector spaces and inner product spaces, as it helps to determine dimensions, linear independence, and the structure of these spaces.

5 Must Know Facts For Your Next Test

1. The span of a single non-zero vector is a line through the origin in the direction of that vector.
2. The span of two non-parallel vectors in two-dimensional space forms a plane that includes all linear combinations of those two vectors.
3. If a set of vectors spans a space, it means every vector in that space can be represented as a combination of those vectors.
4. Vectors are linearly independent if none can be expressed as a linear combination of the others; if they are not independent, their span may be smaller than expected.
5. The span helps to define subspaces, which are essential for understanding concepts such as orthogonality and projections in inner product spaces.

Review Questions

• How does the concept of span relate to the idea of linear independence within a set of vectors?
  • The concept of span is closely tied to linear independence because if a set of vectors is linearly independent, their span will cover an entire subspace without any redundancy. In other words, each vector contributes uniquely to the formation of that subspace. If any vector in the set can be represented as a combination of others, then the span may not fully utilize all dimensions, indicating dependence and limiting its potential coverage.
• Describe how to determine if a set of vectors spans a given vector space, including practical methods or tests.
  • To determine if a set of vectors spans a given vector space, one common method is to form a matrix with these vectors as columns and row-reduce it to echelon form. If the number of leading 1's (pivots) equals the dimension of the space, then the vectors span that space. Another method involves checking if any vector in the space can be written as a linear combination of the given vectors. If this holds true for all possible vectors in the space, then they successfully span it.
• Evaluate how understanding span contributes to deeper insights into inner product spaces and their properties.
  • Understanding span provides critical insights into inner product spaces because it reveals how combinations of vectors interact within these more complex structures. It helps identify orthogonal sets and projections, which are central concepts in inner product spaces. By analyzing spans, one can explore how different vector combinations influence distance and angles between vectors, leading to key applications such as finding orthonormal bases and solving problems related to least squares approximations in high-dimensional spaces.

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.

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 that space can be expressed as a linear combination of the basis vectors.

Dimension:

The dimension of a vector space is the number of vectors in a basis for that space, indicating its size and complexity.