SUMMARY
The vectors [3, -8, 1] and [6, 2, -5] do not form a basis for R3, as they do not span the entire three-dimensional space. However, they are linearly independent and span R2, which is defined as a plane within R3. To establish linear independence, one must verify that neither vector is a scalar multiple of the other and that neither is the zero vector. Thus, these two vectors can be considered a basis for a plane in R3.
PREREQUISITES
- Understanding of linear independence and dependence
- Familiarity with vector spaces and their dimensions
- Knowledge of the concept of spanning sets
- Basic proficiency in linear algebra
NEXT STEPS
- Study the definition and properties of linear independence in vector spaces
- Learn about spanning sets and their significance in linear algebra
- Explore the concept of basis and dimension in Rn
- Investigate examples of vector spans in higher-dimensional spaces
USEFUL FOR
Students of linear algebra, educators teaching vector spaces, and anyone seeking to deepen their understanding of linear independence and spanning sets in R3.