Question

5. Prove that the set of solutions of m linear equations in n variables over a field F is a subspace of FN

Answer 1

Toorove: The set of solutions of m linear equations in n variables.over. a field Os Q : Subspace of_JE: Tonsider the system of linear equations Quitaiz Xiat + Gin 20 = b. axi +922 Kit -- + Ganan be amiru tame X2 t = t. amn 10 = ! can be written as AX= bewhere A an 012 GB - Gin. X- يفيد لية همه ---- Lam am ama amnt. MXn mm Then every Solution (say y Cs whepeed ifiths is a vector in len, CamScanner 2011 -

Thus, the solution set of such a system i*. is a subset of IF. Suppose the system is homogenemas , let the solution set be w s . on set Te KRAX=0. 00: AO=0. Eero vector_(0) EW = W is a non-empty-Subset of IE", Now, suppose u Sv. are two arbitrary elements. of W . Then 4:8 v both are solutions, of AX=0._ - EI. . Auzo and Av: 0 Go for any scalar as b E F (Field), we have S Here we have used Alautby) = a Aut b Avl the property of - - acolt bcol. Linear transformation a autbv is a solution of the homogeneous. _system. AX=0. a autbyt w . & a belf & YVEW.. cs Soannecari's a subspace of len CamScanner

Page ① Result used Suppose w is a subset of a vector T space v. Then w is a subspace L Subspace of rowing two_conditions are satisfieds Tin The zero vector cos belongs to W. for every UVEWALEE the lineup combination au tbv EW:: Note: The solution set w of a non- homogeneous Systems AXE Bi is not a subspace of En. The zero vector. 101 does not belongs to its Solution : seta Remarks I have discussed all the result used while solving the problem : In case you. have any doubt Feel free to ask in the comment boxen cs Scanned with Camscanner