Span

Definition

Span refers to the set of all possible linear combinations of a given set of vectors. It represents all the points that can be reached by scaling and adding these vectors together. Understanding the span is crucial for analyzing the relationships between vectors, determining whether they can cover a certain space, and understanding concepts such as linear independence and basis.

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 linearly independent vectors in three-dimensional space forms a plane through the origin.
3. If the vectors are linearly dependent, their span does not increase beyond the span of any subset of those vectors.
4. To find out if a vector is in the span of a set of vectors, you can set up a system of equations to see if you can express that vector as a linear combination of the others.
5. The concept of span is fundamental in determining whether a set of vectors forms a basis for a vector space.

Review Questions

• How can you determine if a particular vector is included in the span of a given set of vectors?
  • To determine if a vector is included in the span of a set of vectors, you can create a system of linear equations where the target vector is expressed as a linear combination of the given vectors. This involves assigning coefficients to each vector and solving for these coefficients. If you find a solution that satisfies all equations, then the vector lies within the span. If no solution exists, it means that the vector cannot be formed from those vectors.
• In what ways does the concept of span relate to linear independence and basis within a vector space?
  • Span is deeply connected to linear independence and basis because it helps define how much space can be covered by a set of vectors. A basis is defined as a set of vectors that are both linearly independent and whose span fills up the entire vector space. If any vector in a basis can be expressed as a linear combination of others, then it is not independent, which would reduce the dimension and coverage of their span. Thus, analyzing span allows us to understand and verify properties related to independence and basis.
• Evaluate how changing one vector in a linearly independent set affects its span and overall structure as a basis.
  • Changing one vector in a linearly independent set can have significant implications for its span and whether it still forms a basis. If the new vector remains linearly independent from the others, it may expand the span, allowing for more coverage in the vector space. However, if this new vector can be represented as a linear combination of existing vectors, it would render the set linearly dependent, potentially reducing its effectiveness as a basis. This highlights how carefully selecting vectors is critical for maintaining the desired dimensional properties and coverage within the space.

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 linearly independent vectors that spans a vector space, meaning every vector in that space can be expressed as a linear combination of the basis vectors.

Dimension: Dimension refers to the number of vectors in a basis for a vector space, indicating how many degrees of freedom exist within that space.