independence of B, implies ai = bi, for all i. hence uniqueness. Sufficiency To prove B is linearly independent. = = implying ai=0, for each i. hence. Theorem2.21 A linearly independent subset of a finite dimensional vector space can be extended to form a basis of the vector space. Corollary If V is a vector space of dimension n and if X is a linearly independent subset of V with n elements, then X is a basis of V. PRACTICE SUMS Justify whether true/false: (1) every set of three vectors in R2 is linearly dependent, (2) the set {(1,1,0),(0,1,1),(1,0,-1),(1,1,1)} is linearly dependent and one of the vectors can
independence of B, implies ai = bi, for all i. hence uniqueness. Sufficiency To prove B is linearly independent. = = implying ai=0, for each i. hence. Theorem2.21 A linearly independent subset of a finite dimensional vector space can be extended to form a basis of the vector space. Corollary If V is a vector space of dimension n and if X is a linearly independent subset of V with n elements, then X is a basis of V. PRACTICE SUMS Justify whether true/false: (1) every set of three vectors in R2 is linearly dependent, (2) the set {(1,1,0),(0,1,1),(1,0,-1),(1,1,1)} is linearly dependent and one of the vectors can