Question

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.

Forms of linear equations
ANSWERED
asked 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.

Answers (1)

2020-12-16
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 \(9 xx 10\) matrix.
Here, RankA =9 as the system of equation has solution for all \(z in R^9\).
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.
0
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours

Relevant Questions

asked 2020-12-24
Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.
asked 2021-06-10
Determine whether the given set S is a subspace of the vector space V.
A. V=\(P_5\), and S is the subset of \(P_5\) consisting of those polynomials satisfying p(1)>p(0).
B. \(V=R_3\), and S is the set of vectors \((x_1,x_2,x_3)\) in V satisfying \(x_1-6x_2+x_3=5\).
C. \(V=R^n\), and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=\(C^2(I)\), and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.
F. V=\(P_n\), and S is the subset of \(P_n\) consisting of those polynomials satisfying p(0)=0.
G. \(V=M_n(R)\), and S is the subset of all symmetric matrices
asked 2021-06-26
Determine if the statement is true or false, and justify your answer. (a) Different sequence s of row operations can lead to different reduced echelon forms for the same matrix. (b) If a linear system has four equations and seven variables, then it must have infinitely many solutions.
asked 2021-02-11

Let B be a \((4\times3)(4\times3)\) matrix in reduced echelon form.

a) If B has three nonzero rows, then determine the form of B.

b) Suppose that a system of 4 linear equations in 2 unknowns has augmented matrix A, where A is a \((4\times3)(4\times3)\) matrix row equivalent to B.

Demonstrate that the system of equations is inconsistent.

...