Suppose the solutions of a homogeneous system of six linear equations in nine unknowns are all linear combination of four linearly independent nonzero solutions. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.

Accepted Solution

Answer:yes, system have a solutionStep-by-step explanation:Given data linear equation = 6unknown = 9to find outWill the system necessarily have a solutionsolutionwe multiply one non zero solution and we know thatA = 6 × 9 matrixand n = 9so dim (NulA) = 3   because we know (9-6) = 3 and rank of (A) =  n - dim(NulA) = 9 - 3  = 6so we say now here image of A i.e  6 dimensional subspace (A have 6 row)so that Col (A) will be [tex]R^{6}[/tex]so its mean Ax = b has solution when we have bso we now say that yes, system have a solution