Understanding Linear Homogeneous Systems: Finding the Correct Answers

In summary, when a linear homogeneous system Ax=0 has a non-trivial solution and A is an n x n matrix, then the only true statement is that A has rank less than n. Other choices, such as A being invertible, row equivalent to I, or each system Ax=b having a solution, are not necessarily true. It is important to carefully consider the properties of a matrix before choosing the correct answers to these types of questions.
  • #1
DanielJackins
40
0

Homework Statement




If a linear homogeneous system Ax=0 has a non - trivial solution and A is an n x n matrix, then (choose ALL correct answers)

A. A has rank less than n
B. Each system Ax=b with the same coefficient matrix A has a solution
C. A is row equivalent to I
D. If Ax=b has one solution it has many
E. A is invertible

If a linear homogeneous system Ax = 0 has a non - trivial solution and A is n x n, then (choose ALL correct answers)

A. A is invertible
B. A has rank less than n
C. A is row equivalent to I
D. If Ax = b has one solution it has many
E. Each system Ax = b with the same coefficient matrix A has a solution

The Attempt at a Solution



So I looked through my notes for these two questions, found some properties and chose the answers the notes led me to believe were true, but I got both incorrect. I only have one attempt left on each question so I want to make sure I'm 100% sure on the answers Ibefore I try. Could anyone give me a nudge in the right direction? ie. explain linear homogeneous systems?
 
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  • #2
what were the properties that led you to the incorrect answers? What were those answers?

It would help if I knew what I needed to explain.
 
  • #3
My notes said that a square matrix is also invertible, and is row equivalent to I
 
  • #4
Oh, ok. Only a square matrix can be invertible. Not all of them are. Your notes must be mistaken.
 
  • #5
So even though it's a square it isn't necessarily invertible?
 
  • #6
precisely. Read https://www.physicsforums.com/showpost.php?p=2369970&postcount=22" post. I wrote it for somebody that was really being dense so if it seems like it's talking down to you a little, don't take it personally ;)

It gives the simplest examples possible of a matrix equation where the matrix is not invertible. If you have any questions, I'll be happy to answer.
 
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  • #7
Okay thanks for the link. But I'm still kind of lost about the other choices
 
  • #8
The only true statement in both cases is that A has rank less than n.
 
  • #9
There's one more true statement. Which is of course equivalent to the one that you mentioned.
 
  • #10
Sorry what? So A has rank less than n, and there is one more true statement other than that?
 

FAQ: Understanding Linear Homogeneous Systems: Finding the Correct Answers

What is a linear homogeneous system?

A linear homogeneous system is a set of linear equations that can be written in the form Ax = 0, where A is a matrix of coefficients and x is a vector of variables. In other words, all the equations in the system have a solution of x = 0.

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

In a linear homogeneous system, the constant term in each equation is equal to 0. This means that the only solution is x = 0. In a non-homogeneous system, the constant term is not equal to 0, allowing for non-zero solutions.

What is the role of the null space in a linear homogeneous system?

The null space, also known as the kernel, of a linear homogeneous system is the set of all solutions to the system. In other words, it is the set of all possible values for x that satisfy the equations Ax = 0. The null space is important in understanding the behavior of the system and finding the solutions.

Can a linear homogeneous system have more than one solution?

No, a linear homogeneous system can only have one solution which is x = 0. This is because all the equations in the system have a solution of x = 0, making it the only possible solution.

What are some real-world applications of linear homogeneous systems?

Linear homogeneous systems are used in various fields such as physics, engineering, and economics to model and solve problems. For example, in physics, linear homogeneous systems can be used to describe the motion of objects under constant forces. In economics, they can be used to analyze supply and demand equations. Additionally, linear homogeneous systems are also used in computer graphics and image processing to transform and manipulate images.

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