Linear Dependence with Zero Vector: A Simple Solution?

Thread starter eyehategod
Start date Nov 4, 2007
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Algebra Linear Linear algebra

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SUMMARY

A set containing the zero vector is always linearly dependent. This is because the definition of linear dependence states that a set of vectors is linearly dependent if there exist constants, not all zero, such that their linear combination equals the zero vector. In the case of a single vector, the zero vector itself can be represented as a linear combination with the constant being non-zero, thus confirming the dependence.

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Understanding of linear algebra concepts, specifically linear dependence
Familiarity with vector spaces and their properties
Knowledge of linear combinations and scalar multiplication
Basic grasp of mathematical proofs and logic

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Study the properties of vector spaces in linear algebra
Learn about the implications of linear independence versus linear dependence
Explore examples of linear combinations in various vector spaces
Investigate the role of the zero vector in higher-dimensional spaces

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Students of linear algebra, educators teaching vector space concepts, and anyone interested in mathematical proofs related to linear dependence.

Nov 4, 2007

eyehategod

I have to prove that a set that has the zero vector is linearly dependent. Can anyone help me out?

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Nov 4, 2007

Dick

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Homework Helper

As usual a set {x1,x2...} is linearly dependent if you can find constants (not all zero) such that c1*x1+c2*x2+...=0. What happens if there is only one vector? Doesn't that make it easy?