SUMMARY
If a set of vectors {x1, x2, ..., xn} spans the vector space $\mathbb{R}^n$, then these vectors are linearly independent. This conclusion arises from the fact that fewer than n vectors cannot span $\mathbb{R}^n$. Therefore, a set of n vectors that spans $\mathbb{R}^n$ forms a basis for this space, confirming their linear independence.
PREREQUISITES
- Understanding of vector spaces, specifically $\mathbb{R}^n$
- Knowledge of linear independence and spanning sets
- Familiarity with the concept of basis in linear algebra
- Basic proficiency in mathematical notation and terminology
NEXT STEPS
- Study the properties of vector spaces in linear algebra
- Learn about the criteria for linear independence of vectors
- Explore the concept of bases and dimension in vector spaces
- Investigate applications of linear independence in solving linear equations
USEFUL FOR
Students of linear algebra, mathematicians, and educators seeking to deepen their understanding of vector spaces and linear independence.