SUMMARY
The solution space W of the matrix A, defined as W = {x ∈ R^3 | Ax = 0}, is determined through the row-reduced echelon form (RREF) of A, which is given as [1 0 5], [0 1 1], [0 0 0]. To prove that W is a subspace of R^3, one must verify that W satisfies the vector space axioms, specifically closure under addition and scalar multiplication. The RREF indicates that W is spanned by the vectors derived from the free variables in the system, confirming its subspace status.
PREREQUISITES
- Understanding of linear algebra concepts, specifically solution spaces.
- Familiarity with row-reduced echelon form (RREF) and its significance.
- Knowledge of vector space axioms and their application.
- Proficiency in solving homogeneous systems of linear equations.
NEXT STEPS
- Study the properties of vector spaces and subspaces in linear algebra.
- Learn how to derive the general solution from RREF matrices.
- Explore examples of proving subspace properties using vector space axioms.
- Investigate the implications of free variables in the context of solution spaces.
USEFUL FOR
Students of linear algebra, educators teaching matrix theory, and anyone seeking to deepen their understanding of vector spaces and subspace proofs.