SUMMARY
In a vector space V with three linearly independent vectors e1, e2, and e3, the vectors e1+e2, e2-e3, and e3+2e1 are also proven to be linearly independent. This is established by analyzing the linear combination c1(e1+e2) + c2(e2-e3) + c3(e3+2e1) = 0, which leads to the rearranged equation (c1+2c3)e1 + (c1+c2)e2 + (c3-c2)e3 = 0. The independence of e1, e2, and e3 implies that the coefficients c1+2c3, c1+c2, and c3-c2 must all equal zero, confirming the linear independence of the new vectors.
PREREQUISITES
- Understanding of vector spaces and linear independence
- Familiarity with linear combinations of vectors
- Knowledge of coefficient comparison in vector equations
- Basic proficiency in mathematical notation and proofs
NEXT STEPS
- Study the concept of basis in vector spaces
- Learn about the Rank-Nullity Theorem in linear algebra
- Explore applications of linear independence in higher dimensions
- Investigate the implications of linear transformations on vector independence
USEFUL FOR
Students and educators in mathematics, particularly those focusing on linear algebra, as well as researchers needing to understand vector independence in theoretical contexts.