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You have to keep refreshing the page.Brian_D said:
The vectors should be columns in your matrix. It doesn't work if you make the vectors the rows. You should check this yourself.Brian_D said:
Conflicting definitions of linear independence
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SUMMARY
This discussion clarifies the definitions of linear independence and dependence in the context of a set of vectors. A set of vectors is linearly independent if the only solution to the homogeneous equation is the trivial solution, which occurs when the coefficients of the vectors are all zero. However, if a set contains a zero vector or is underdetermined, it is considered linearly dependent. The confusion arises when analyzing a homogeneous system with a zero column, which implies infinitely many solutions, thus indicating dependence despite the initial claim of independence based on the first three vectors.
PREREQUISITES- Understanding of linear algebra concepts, particularly linear independence and dependence.
- Familiarity with homogeneous systems of equations.
- Knowledge of matrix rank and its implications on vector independence.
- Basic proficiency in Gaussian elimination for solving linear systems.
- Study the concept of matrix rank and its role in determining linear independence.
- Learn about Gaussian elimination and how it applies to solving homogeneous systems.
- Explore the implications of zero vectors in linear algebra and their effect on vector sets.
- Investigate the differences between independent equations and linearly independent vectors.
Students of linear algebra, educators teaching vector spaces, and anyone seeking to clarify the concepts of linear independence and dependence in mathematical contexts.
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