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.

Question
Forms of linear equations
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.

Answers (1)

2020-12-25
Consider the provided question,
Given a system of five linear equations in six unknowns are all multiples of one nonzero solution.
It can be written as,
Ax=0 where A=5*6 matrix
dim(NulA)=1 because all solutions of Ax=0 are multiples of nonzero solution
We have to explain that the system necessarily have a solution for every possible choice of constants on the right sides of the equations.
Since,A is 5*6 matrix n=6
rank(A)=n-dim(NulA)=6-1=5
Image of A is 5 dimensional subspace of \(R^5\) (because A has 5 rows)
So,Col(A)=\(R^5\)
This means that Ax=b has a solution for every b
0

Relevant Questions

asked 2021-03-04
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?
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.
asked 2020-12-15
Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent?
asked 2020-11-08
Suppose that a nonhomogeneous system with 10 linear equations in 8 unknowns has a solution with 2 free variables. Is it possible to change some constants on the equations’ right hand side to make the new system inconsistent?
asked 2020-10-21
Given a homogeneous system of linear equations, if the system is overdetermined, what are the possibilities as to the number of solutions?
Explain.
asked 2021-01-04
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
asked 2021-02-11
Let B be a (4×3)(4×3) 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 (4×3)(4×3) matrix row equivalent to B. Demonstrate that the system of equations is inconsistent.
asked 2021-02-08
Find values of a and b such that the system of linear equations has (a) no solution, (b) exactly one solution, and (c) infinitely many solutions.
x + 2y = 3
ax + by = −9
asked 2021-02-25
Each equation in a system of linear equations has infinitely many ordered-pair solutions.Determine whether the statement makes sense or does not make sense, and explain your reasoning.
asked 2021-02-12
Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. All vectors (x, y, z) in \(V_3\) whose components satisfy a system of three linear equations of the form:
\(a_{11}x+a_{12}y+a_{13}z=0\)
\(a_{21}x+a_{22}y+a_{23}z=0\)
\(a_{31}x+a_{32}y+a_{33}z=0\)
...