Linearly dependent

Definition

Linearly dependent refers to a set of vectors in which at least one vector can be expressed as a linear combination of the others. This means that there exists a non-trivial solution to the equation formed by setting the linear combination of these vectors equal to zero. In this context, understanding linear dependence is crucial for analyzing the relationships between vectors and determining their independence or redundancy in spanning a vector space.

5 Must Know Facts For Your Next Test

1. If a set of vectors is linearly dependent, it means that at least one vector can be written as a combination of others, indicating redundancy in the set.
2. The presence of a zero vector within any set of vectors automatically makes the set linearly dependent.
3. For any set of more than 'n' vectors in an 'n'-dimensional space, the set must be linearly dependent due to the inability to have more unique directions than dimensions.
4. Linearly dependent sets do not span the entire vector space because they contain redundant information.
5. Checking for linear dependence often involves setting up a matrix and finding whether the determinant equals zero or if there are free variables in a system of equations.

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, you can create a matrix with those vectors as columns and perform row reduction to check for pivot positions. If there are fewer pivot columns than the number of vectors, it indicates that at least one vector can be written as a combination of others, confirming linear dependence. Alternatively, if the determinant of the matrix formed by these vectors equals zero, this also indicates linear dependence.
• Discuss the implications of having a linearly dependent set of vectors in terms of spanning and dimension.
  • A linearly dependent set of vectors does not span the entire vector space since there are redundancies present within the vectors. This means that there is not enough unique directionality to cover all dimensions effectively. For example, in a 3-dimensional space, having four vectors would imply that at least one vector is unnecessary and does not contribute additional information. Consequently, the effective dimension spanned by such a set would be less than three.
• Evaluate how linear dependence relates to solving systems of equations and the uniqueness of solutions.
  • In systems of equations represented by matrices, linear dependence among rows or columns indicates that some equations are redundant. This can lead to either no solutions or infinitely many solutions instead of a unique solution. When a system has fewer independent equations than unknowns due to linear dependence, it often results in free variables that provide multiple valid solutions. Thus, understanding linear dependence is essential for determining solution characteristics and system behavior.

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 points that can be reached using those vectors.

Linearly Independent: A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others, indicating that each vector adds unique information.