SUMMARY
The vectors v1 = [1, -3], v2 = [2, -8], and v3 = [-3, 7] span R² but do not form a basis due to linear dependence, specifically v3 = v1 + v2. To express the vector [1, 1] as a linear combination, two valid representations are 5v1 - 2v2 = [1, 1] and 10v1 - 3v2 + v3 = [1, 1]. The first representation arises from solving the system with v1 and v2, while the second incorporates the linearly dependent vector v3, leading to an infinite number of solutions.
PREREQUISITES
- Understanding of linear combinations
- Knowledge of vector spaces and spanning sets
- Familiarity with linear dependence and independence
- Basic skills in solving systems of linear equations
NEXT STEPS
- Study the concept of linear combinations in depth
- Learn about vector spaces and their properties
- Explore methods for determining linear independence
- Practice solving systems of linear equations with multiple solutions
USEFUL FOR
Students studying linear algebra, educators teaching vector spaces, and anyone interested in understanding linear combinations and their applications in R².