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

Clifland

Answered question

2020-12-15

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.

Answer & Explanation

Fatema Sutton

Fatema Sutton

Skilled2020-12-16Added 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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?