SUMMARY
The discussion centers on proving that the set (u, v, u+v) cannot form a basis for the vector space spanned by . The proof demonstrates linear dependence by showing that the equation αu + βv + γ(u+v) = 0 has non-trivial solutions for α, β, and γ. Specifically, the values α = -1, β = -1, and γ = 1 satisfy this equation, confirming that the vectors are linearly dependent and thus do not form a basis.
PREREQUISITES
- Understanding of vector spaces and bases
- Knowledge of linear dependence and independence
- Familiarity with linear combinations of vectors
- Basic algebraic manipulation skills
NEXT STEPS
- Study the concept of vector spaces in linear algebra
- Learn about the criteria for linear independence
- Explore examples of bases in different vector spaces
- Practice solving linear equations involving multiple variables
USEFUL FOR
Students of linear algebra, mathematics educators, and anyone interested in understanding vector spaces and linear dependence.