SUMMARY
The discussion centers on finding a vector w in R3 that is not a linear combination of the vectors v1 = [1;2;-1] and v2 = [2;-1;-2]. It is established that any vector not within the span of v1 and v2 qualifies, with the cross product of v1 and v2 serving as a definitive example. The cross product yields a vector orthogonal to both v1 and v2, ensuring it is not a linear combination of these vectors. The chosen vector [1; 1; 1] is also valid as it lies outside the span of v1 and v2.
PREREQUISITES
- Understanding of vector spaces and linear combinations
- Familiarity with the concept of span in linear algebra
- Knowledge of the cross product in R3
- Basic proficiency in matrix notation and operations
NEXT STEPS
- Study the properties of vector spans and linear independence
- Learn how to compute the cross product of two vectors in R3
- Explore examples of vectors that are not linear combinations of given vectors
- Investigate applications of orthogonal vectors in linear algebra
USEFUL FOR
Students and educators in linear algebra, mathematicians exploring vector spaces, and anyone seeking to understand the concepts of linear combinations and orthogonality in R3.