What does a family of nontrivial coefficients prove about linear dependence?
Conditions
- You have a specific set of vectors.
- You have found a parametric family of solutions to the homogeneous equation.
- The parameter can be chosen to be nonzero.
Reasoning, step by step
- Identify the coefficients from the solution to the homogeneous system.
- Verify that at least one coefficient can be nonzero (by choosing a nonzero parameter).
- Confirm that the linear combination of the vectors with these coefficients equals the zero vector.
- Conclude that the existence of such nontrivial coefficients proves linear dependence.
Example
The script states: 'The example gives coefficients (−,−,k). Taking k nonzero produces a zero vector combination with nonzero coefficients, so the vectors are dependent.'
Common misconceptions
- Thinking that the coefficients must be integers.
- Believing that a single nonzero coefficient is enough without checking the combination equals zero.
- Confusing the parameter with the vectors themselves.
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Conditions: The set contains at least two vectors.; One vector lies in the span of the other vectors.
A set of vectors is linearly independent exactly when the only linear combination that equals the zero vector forces all coefficients to be zero. If there exists a nontrivial coefficient choice (at least one coefficient is nonzero) that produces the zero vector, the set is linearly dependent.
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