Solution to system of linear equations in range of system matrix

In summary, the conversation discusses a problem with finding solutions for three systems, where solutions exist for systems 1 and 2 but not for 3. The person is trying to find a difference between the matrices of systems 2 and 3 that could be linked to the issue, and is given a hint to consider whether a certain equation has solutions. They are also directed to a Wikipedia entry for more information.
  • #1
kalleC
11
0

Homework Statement


See image. a) and b) have been solved. The problem is c)


Homework Equations





The Attempt at a Solution


I really have no idea where to begin. For the three systems given there are solutions x in range(A) for system 1 and 2 but not for 3. Therefore I have been trying to spot some obvious difference between system matrix A of system 2 and 3 but I cannot think of any clearly different property that could be linked to b.

Any hint just to get me started would be much appreciated.
 

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  • #2
Hint: Ask yourself whether [itex] A^T y = 0 [/tex] has solutions such that [itex] y^T b \neq 0[/itex].

Try it yourself first. Then if you think you know what is going on, look at the wikipedia entry under Fredholm alternative.
 
  • #3
Much appreciated!
 

1. What is a system of linear equations?

A system of linear equations is a group of two or more equations that contain two or more variables. The solution to a system of linear equations is the set of values that satisfy all of the equations in the system.

2. What is the range of a system matrix?

The range of a system matrix is the set of all possible outputs that can be generated by the system. In other words, it is the span of the columns of the matrix, or the set of all linear combinations of the columns.

3. How do you find the solution to a system of linear equations in the range of the system matrix?

To find the solution, you can use a variety of methods such as Gaussian elimination, Cramer's rule, or matrix inversion. These methods involve manipulating the system of equations and using basic algebra to solve for the variables.

4. Can a system of linear equations have more than one solution in the range of the system matrix?

Yes, a system of linear equations can have infinitely many solutions in the range of the system matrix. This occurs when the system of equations is consistent and there are more variables than equations, allowing for multiple possible combinations of values.

5. What is the importance of finding the solution to a system of linear equations in the range of the system matrix?

Finding the solution to a system of linear equations in the range of the system matrix can provide valuable information about the relationships between the variables in the system. It can also be used to solve real-world problems involving multiple variables and equations, such as in economics, engineering, and science.

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