Let F be any field. Consider the "system of m simultaneous linear equations inn unknowns" 

where a_ij, b_i ∈ F.

a. Show that the "system has a solution;' that is, there exist X₁, ··· , X_n ∈ F that satisfy all m equations, if and only if the vector β = (b₁ , ··· , b_m) of F^m is a linear combination of the vectors α_j = ( a₁j , ··· , a_mj). (This result is straightforward to prove, being practically the definition of a solution, but should really be regarded as the fundamental existence theorem for a simultaneous solution of a system of linear equations.) 

b. From part (a), show that if n = m and {α_j |j = 1, ··· , n} is a basis for Fⁿ, then the system always has a unique solution.

Transcribed Image Text:

a₁1X₁ + a12X₂ + ... + a₁n Xn +ain Xn = b₁, a21 X₁ + a22X2 + ... + a2nXn=b2, am1X1 + am2X2 + +amn Xn = bm,