Question

Determine whether S = { (1,0,0) , (1,1,0) , (1,1,1) } Is a basis for R^3 and write (8,3,8) as a linear combination of vector in S.

Please explain in details how to solve this problem.

Answer 1

T1Ven Bv definiti on Aset Bis a basis for a vector space Vif B is linearly independent and it spans V. We say that the vectors {y,va».., Jare linearly dependent if there exists scalars a, a not all of them zero such that CvCvcv0 By definition, we say that the vectors0%½, ,y) are linearly dependent if there exists scalars a, a not all of them zero such that ay+a2v, + +4", =0 We say that the vectors (vi,v.., vt) are linearly independent ,a 4 Consi der Writting in the matrixform AX-0,weget 0 11is in echelon form A and A0 are in echelon form AlsoRank(A) Rank(A10)-number of non-zerorows inits echelon form-no of unknowns 3 The given set of vectors is linearly independent By back substitution, we get a=b=c=0 u, v, w| is linearly independent

To show that Sspans IR a+b+c (x c = z → (iii) By abck substitution, Consi der bte-y Consider a +0+c= x Sspans IR3 is a basis of R Now 3(8-3) 0(3-8)1(81(5)51+(8)1 3(5) 05(8)1,is the linear combination of |3 w.rt. the given basis S