Linear System Solutions and the Role of Scalar Multiplication

In summary, the statement suggests that if the vectors of v and u are solutions of the nonhomogeneous linear system Ax=b, then any linear combination of these vectors (ru+sv) will also be a solution for any real values of r and s. This statement is true for nonhomogeneous systems and can be used to answer both questions. However, a counterexample can be easily found for the first question, and for the second question, one can try writing the matrix equation for Ax=b using ru+sv.
  • #1
golriz
43
0
I have a question:
If the vectors of v & u are solutions of the nonhomogeneous linear system Ax=b, then ru+sv is a solution of the nonhomogeneous system for any real values of r and s.
is this statement true?
is this statement true for homogeneous systems too?
 
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  • #2
Hint: When a nonhomogeneous linear system has infinite solutions, then the general solutions can be written in parametric vector form.

This can be used to answer both questions.
 
  • #3
You should be able to find a counterexample easily for the first question. For the second, try writing the matrix equation for Ax=b using ru+sv.
 

FAQ: Linear System Solutions and the Role of Scalar Multiplication

What is a homogeneous linear system?

A homogeneous linear system is a set of linear equations where all the terms have a degree of 1 and the constant term is equal to 0. This means that all the equations in the system have the same variables and they are all equal to 0 when solved.

What is the solution to a homogeneous linear system?

The solution to a homogeneous linear system is either a unique solution where all the variables are equal to 0, or an infinite number of solutions where the variables can take on any value. This is because the equations in a homogeneous linear system are all consistent and can be solved simultaneously.

How do you solve a homogeneous linear system?

To solve a homogeneous linear system, you can use methods such as Gaussian elimination, Cramer's rule, or matrix inversion. These methods involve manipulating the equations to eliminate variables and find the values of the remaining variables.

What are the applications of homogeneous linear systems?

Homogeneous linear systems are commonly used in various fields of science, such as physics, engineering, and economics. They can be used to model and solve problems involving linear relationships, such as calculating forces in a mechanical system or predicting the behavior of a chemical reaction.

What is the difference between a homogeneous and non-homogeneous linear system?

The main difference between a homogeneous and non-homogeneous linear system is the constant term. In a homogeneous linear system, the constant term is equal to 0, while in a non-homogeneous linear system, it can take on any value. This means that the solution to a non-homogeneous linear system will always have a unique solution, whereas a homogeneous linear system may have either a unique solution or an infinite number of solutions.

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