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Prove that two vectors are linearly dependent (taken as a set of vectors) if and only if one vector is multiple of the other.

Prove that two vectors are linearly dependent (taken as a set of vectors) if and only if one vector is multiple of the other.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Prove that two vectors are linearly dependent (taken as a set of vectors) if and only if one vector is a multiple of the other.** This statement explores the concept of linear dependence in vector spaces. If two vectors are linearly dependent, it means that they lie along the same line in a vector space, implying that one is a scalar multiple of the other. Essentially, neither vector adds a new dimension to the span of the set, leading to a dependency between them. Understanding this relationship is crucial in linear algebra, as it forms the basis for more complex analyses of vector spaces and their dimensions.
Transcribed Image Text:**Prove that two vectors are linearly dependent (taken as a set of vectors) if and only if one vector is a multiple of the other.** This statement explores the concept of linear dependence in vector spaces. If two vectors are linearly dependent, it means that they lie along the same line in a vector space, implying that one is a scalar multiple of the other. Essentially, neither vector adds a new dimension to the span of the set, leading to a dependency between them. Understanding this relationship is crucial in linear algebra, as it forms the basis for more complex analyses of vector spaces and their dimensions.
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