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 helps determine the extent to which a set of vectors can cover a vector space, indicating whether the vectors can be combined to produce other vectors within that space. Understanding the span is crucial for analyzing linear systems, as it relates to the solutions of equations and their representation in two and three dimensions.

5 Must Know Facts For Your Next Test

1. The span of a single non-zero vector in two or three dimensions forms a line through the origin.
2. The span of two non-parallel vectors in two dimensions creates a plane, while in three dimensions, it can form a plane or space depending on their linear independence.
3. If a set of vectors spans a vector space, any vector within that space can be expressed as a linear combination of those vectors.
4. In linear systems, determining the span helps identify whether the system has no solution, one solution, or infinitely many solutions based on the dimensions involved.
5. The concept of span is closely related to the rank of a matrix, which indicates the maximum number of linearly independent column vectors in the matrix.

Review Questions

• How does understanding the span of a set of vectors help in solving linear systems?
  • Understanding the span of a set of vectors is essential when solving linear systems because it reveals if the equations represented by those vectors have solutions. If the span covers the entire space relevant to the system, it indicates that any vector in that space can be formed from linear combinations of the spanning vectors. Conversely, if the span does not cover the required space, it suggests that some solutions may be unattainable, leading to either no solutions or infinitely many solutions depending on how the equations relate.
• Compare and contrast the span generated by one vector versus two vectors in two-dimensional space.
  • The span generated by one non-zero vector in two-dimensional space results in a line through the origin, which means all points along that line can be represented as multiples of that vector. In contrast, when two non-parallel vectors are involved, their span forms an entire plane through the origin. This plane includes all linear combinations of the two vectors, demonstrating how multiple directions (or degrees of freedom) allow for covering more area compared to just one direction with a single vector.
• Evaluate how changing the number or nature of vectors affects the span and implications for linear systems.
  • Changing the number or nature of vectors can significantly affect their span and, consequently, the implications for linear systems. For instance, adding more linearly independent vectors increases dimensionality and expands coverage in space. However, if new vectors are dependent on existing ones (i.e., can be expressed as combinations of them), they do not add to the span. This situation can impact solutions to linear equations; more independent vectors generally lead to better possibilities for finding solutions, while dependence may indicate redundancy or constraints in forming all necessary combinations for valid solutions.

Related terms

Linear Combination: A linear combination of vectors involves multiplying each vector by a scalar and adding the results together.

Vector Space:

A vector space is a collection of vectors that can be added together and multiplied by scalars, adhering to certain rules and properties.

Basis: A basis is a set of vectors in a vector space that is linearly independent and spans the entire space.