Vector Space Solutions for Systems: Explained Here

In summary, the conversation discusses whether a solution set for a particular system is a vector space. It is generally not true, as vector spaces are closed under scalar multiplication and the set of solutions may not be. If there is a unique solution, then the solution set would only have one element, making it a degenerate vector space. However, in the case of a system of homogeneous equations, the solution set is a subspace. If the system has the same number of independent equations as unknowns, the solution set is the zero vector, but if the rank is lower, it is a non-trivial subspace of R[sup]n[/sub].
  • #1
EvLer
458
0
Hi everyone,
general question: is a solution set for a particular system a vector space? I know it can be if there is a unique solution, but is it generally true?
Could someone explain, please?

Thanks.
 
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  • #2
No. Vector spaces are closed under scalar multiplication. If b is a scalar not equal to 1, Y is non-zero, and X is a solution of AX = Y, then:

A(bX) = b(AX) = bY is not equal to Y, so (bX) is not a solution, so the set of solutions is not closed under scalar multiplication, so the set of solutions is not a vector space. Perhaps I've misinterpreted your question. If there is a unique solution, then there would only be that 1 element of the vector space. The only vector space that has only one element is the degenerate vector space {0}.
 
  • #3
For a particular system? Do you mean a system of linear equations?

The solution set of a system of homogenous equations is a subspace.

If the system consists of n independent equations in n unknowns, then it is just the 0 vector but if the rank is lower than the number of unknowns, then it is a non-trivial subspace of Rn[/sub].
 

1. What is a vector space?

A vector space is a mathematical concept that describes a set of objects called vectors, which can be added together and multiplied by numbers to produce new vectors. It is a fundamental tool in linear algebra and is used to study systems of equations and other mathematical structures.

2. How are vector spaces used in systems?

Vector spaces are used to represent and solve linear systems of equations. The vectors in the vector space correspond to the unknown variables in the system, and the operations of addition and scalar multiplication correspond to the equations and their solutions.

3. Can vector spaces be used for non-linear systems?

No, vector spaces are specifically designed for linear systems. Non-linear systems require different mathematical tools and concepts for analysis and solution.

4. What are some real-life applications of vector space solutions for systems?

Vector space solutions for systems have many practical applications in fields such as engineering, physics, economics, and computer science. They can be used to model and solve problems involving multiple variables and equations, such as circuit analysis, optimization, and data analysis.

5. Are there any limitations to using vector space solutions for systems?

While vector spaces are powerful tools for solving linear systems, they do have limitations. They cannot be used for non-linear systems and may not always provide unique solutions. Additionally, they may not be applicable to all real-world problems and may require simplifications and assumptions to be made.

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