Linear Combination

Definition

A linear combination is an expression formed by multiplying each vector in a set by a corresponding scalar and then summing the results. This concept is essential for understanding how vectors can be combined to produce new vectors and plays a crucial role in defining vector spaces, determining the structure of subspaces, and assessing linear independence or dependence among vectors.

5 Must Know Facts For Your Next Test

1. Every vector in a vector space can be expressed as a linear combination of its basis vectors.
2. If a set of vectors can be expressed as a linear combination of other vectors, they are said to be linearly dependent.
3. The coefficients used in a linear combination are often referred to as scalars and can come from any field, such as real or complex numbers.
4. The concept of linear combinations extends to infinite-dimensional spaces, where an infinite number of basis vectors may be required to express other vectors.
5. Determining whether a set of vectors is linearly independent or dependent often involves checking if the only solution to their linear combination equating to zero is the trivial solution.

Review Questions

• How does the concept of linear combination relate to the idea of vector spaces and their structure?
  • Linear combinations are foundational to vector spaces because they allow us to form new vectors from existing ones. In any vector space, any vector can be expressed as a linear combination of basis vectors. This relationship ensures that the space is closed under addition and scalar multiplication, adhering to the properties that define a vector space.
• Discuss how linear combinations help identify the sum and direct sum of subspaces.
  • Linear combinations play a crucial role in determining the sum and direct sum of subspaces. When combining two subspaces, their sum consists of all possible linear combinations formed by taking one vector from each subspace. If the only way to express the zero vector as a linear combination from both subspaces is by using zero vectors from each, then the sum is a direct sum, indicating that they intersect trivially.
• Evaluate the importance of linear combinations in establishing the concepts of linear independence and dependence among vectors.
  • Linear combinations are central to understanding linear independence and dependence. A set of vectors is linearly independent if the only way to express the zero vector as a linear combination is by using all zero coefficients. If there exists a non-trivial linear combination where at least one coefficient is non-zero yielding the zero vector, then those vectors are linearly dependent. This assessment helps determine which sets of vectors can serve as bases for vector spaces.

Related terms

Vector Space: A collection of vectors that can be added together and multiplied by scalars, satisfying certain axioms like closure, associativity, and the existence of an additive identity.

Span: The set of all possible linear combinations of a given set of vectors, effectively describing all the vectors that can be generated from those original vectors.

Basis: A set of linearly independent vectors that span a vector space, serving as the building blocks for expressing any vector within that space as a linear combination.