Span

Definition

In linear algebra, the span of a set of vectors is the collection of all possible linear combinations of those vectors. It reflects the idea that by taking different multiples of these vectors and adding them together, you can create a whole range of new vectors, effectively filling a certain space in vector space. Understanding span is essential for grasping concepts like vector spaces and linear independence, as it helps determine the dimensionality of the space formed by a set of vectors.

5 Must Know Facts For Your Next Test

1. The span of a single non-zero vector is a line in the direction of that vector, while the span of two linearly independent vectors in two-dimensional space forms a plane.
2. If a set of vectors spans a vector space, then any vector in that space can be expressed as a linear combination of those spanning vectors.
3. If a set of vectors spans a space but contains redundant vectors (linearly dependent), then removing those redundant vectors will still span the same space.
4. The dimension of the span corresponds to the maximum number of linearly independent vectors that can be formed from the original set.
5. To determine if a set of vectors spans a particular space, you can check if the system of equations formed by setting them equal to any vector in that space has a solution.

Review Questions

• How does the concept of span relate to understanding whether a set of vectors is linearly independent?
  • The span helps in understanding linear independence because if the span of a set of vectors equals the entire vector space, it indicates that those vectors can express every vector within that space. However, if some vectors are linearly dependent, their span might not cover the entire space unless combined with additional independent vectors. Thus, analyzing the span gives insight into both the dimensionality and independence of the given set.
• What is the relationship between the span of a set of vectors and the dimension of the vector space they occupy?
  • The relationship between span and dimension is that the dimension of the span corresponds to how many linearly independent vectors are included in that set. For instance, if three vectors are in three-dimensional space but only two are independent, the span will create a plane rather than fill the whole three-dimensional space. This means that knowing how many independent vectors are present directly informs you about the dimensionality represented by their span.
• Evaluate how altering one vector in a set affects its overall span and implications for linear combinations.
  • Altering one vector in a set can significantly change its overall span, particularly if the new vector introduces linear independence or dependence. If replaced by another dependent vector, it may not expand the span beyond what was already achieved. Conversely, introducing an independent vector could increase the dimensionality of the span, allowing for more diverse linear combinations. This evaluation shows how delicate and interconnected these relationships are within linear algebra.

Related terms

Linear Combination: A linear combination involves multiplying each vector in a set by a scalar and then adding the results together to form a new vector.

Vector Space:

A vector space is a collection of vectors that can be added together and multiplied by scalars while satisfying certain axioms like closure and associativity.

Basis: A basis is a set of linearly independent vectors in a vector space that can be combined to form any vector in that space through linear combinations.