How to show vectors are linearly independent?
Understand the Problem
The question is asking for methods or techniques to demonstrate that a set of vectors does not depend linearly on each other. This typically involves showing that the only solution to a linear combination of the vectors equating to zero is the trivial solution where all coefficients are zero.
Answer
Calculate the determinant of the matrix formed by the vectors.
The final answer is to check the matrix formed by the vectors and compute its determinant.
Answer for screen readers
The final answer is to check the matrix formed by the vectors and compute its determinant.
More Information
If the vectors form a square matrix and the determinant is non-zero, the vectors are guaranteed to be linearly independent.
Tips
A common mistake is forgetting that this method only works if the number of vectors equals their dimension (i.e., a square matrix).
Sources
- Determine if vectors are linearly independent - math.stackexchange.com
- Span, linear independence and basis Rank and nullity - CSI Math - math.csi.cuny.edu
- Testing for Linear Dependence of Vectors - Oregon State University - sites.science.oregonstate.edu
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