Suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss.

Clifland 2020-12-15 Answered
Suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss.
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Expert Answer

Fatema Sutton
Answered 2020-12-16 Author has 88 answers

To check:
Whether it is possible to find two non-zero solutions of the associated homogenous m use system that are not the multiple of each other.
Theorem used:
The dimension of the column space and the row space of an mxn matrix A are equal. This common dimension, the rank of A, also equals to the number of pivot positions in A and satisfies the equation Rank(A)+dim Nul(A)=n.
Explanation: It is given that a non-homogeneous system of 9 linear equations and 10 unknowns has a solution for all possible constants on the right hand side of the equation.
Consider the system of equation Ax=z
The matrix A of order m*nrepresents it contain m rows and n columns.
Here, A is a 9×10 matrix.
Here, RankA =9 as the system of equation has solution for all zinR9.
Rank(A)+ dim Nul(A) =n dim(nul4)+RankA =10
dim(nul4) = 10-9
dim(nulA) = 1
That is, it is impossible to find two linearly independent vectors in NullA.

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