written as a linear combination of the others Definition The vector V1, V2, . Vn in a vector space V are said to be linearly independent if and only if the solution to C1V1+C2V2+ The vector d 2 are linear independent, since if C +c2 0 C1+2c2=0 and the only solution to this system is 3.4 Subspace Definition The set of all linear combinations of a set of vectors is the vector space that is spanned by those vectors Example R=Spn(v1…,k) for a basis(v1,…,vk) We now consider what happens to the vector space that is spanned by linearly dependent vectorswritten as a linear combination of the others. Definition: The vector v1, v2, ..., vn in a vector space V are said to be linearly independent if and only if the solution to c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0. Example: The vector  1 1  and  1 2  are linear independent, since if c1  1 1  + c2  1 2  =  0 0  , then c1 + c2 = 0 c1 + 2c2 = 0 and the only solution to this system is c1 = c2 = 0. 3.4 Subspace Definition: The set of all linear combinations of a set of vectors is the vector space that is spanned by those vectors. Example: R k = Span(v1, ..., vk) for a basis (v1, ..., vk). We now consider what happens to the vector space that is spanned by linearly dependent vectors. 10