Linearly dependent

Definition

Linearly dependent refers to a situation in which a set of vectors is such that at least one vector can be expressed as a linear combination of the others. This concept is fundamental in understanding the structure of vector spaces, as it indicates redundancy among the vectors in the set. When vectors are linearly dependent, it signifies that they do not all contribute unique directions in the space, which is crucial for spanning a vector space effectively.

5 Must Know Facts For Your Next Test

1. If a set of vectors is linearly dependent, at least one vector in that set can be written as a combination of the others, meaning they do not all point in unique directions.
2. In practical terms, if there are more vectors than the dimension of the vector space, the set must be linearly dependent.
3. The presence of a zero vector in any set of vectors automatically indicates linear dependence since it can be expressed as a linear combination of itself with any scalar multiplier.
4. Checking for linear dependence can often be done using techniques such as row reduction on a matrix formed by the vectors or calculating the determinant of a matrix if there are as many vectors as dimensions.
5. Linearly dependent sets have implications for solving systems of equations, as they can lead to either no solutions or infinitely many solutions depending on their relationships.

Review Questions

• How can you determine if a set of vectors is linearly dependent or independent?
  • To determine if a set of vectors is linearly dependent or independent, you can create a matrix using these vectors as columns and perform row reduction to echelon form. If you find any row that consists entirely of zeros, or if there are fewer pivot positions than there are vectors, this indicates that the vectors are linearly dependent. Additionally, if any vector can be expressed as a linear combination of others, that confirms dependence.
• Discuss the significance of linear dependence in the context of spanning sets and bases.
  • Linear dependence plays a crucial role in understanding spanning sets and bases within vector spaces. A spanning set may include linearly dependent vectors, but to form a basis for the vector space, all vectors must be linearly independent. This means that each vector contributes uniquely to the span; if any vector is dependent on others, it can be removed without losing any span information. Therefore, identifying linear dependence helps in refining sets to find minimal bases.
• Evaluate how linear dependence among vectors affects solutions to linear equations and vector space properties.
  • Linear dependence among vectors significantly impacts the solution structure of systems of linear equations. If the system's coefficient matrix has linearly dependent columns, this often results in either no solution or infinitely many solutions rather than a unique solution. This is tied to properties such as dimension and rank; specifically, if there are dependencies among vectors representing equations, it may lead to underdetermined systems where one or more equations do not add new information. Thus, understanding linear dependence is key for solving problems in higher dimensions effectively.

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.

Span: The span of a set of vectors is the set of all possible linear combinations of those vectors, representing all the points that can be reached by those vectors in the vector space.

Basis: A basis is a set of linearly independent vectors that span a vector space, serving as a minimal set needed to describe the entire space.