LINEAR ALGEBRA: nontrivial solutions

In summary, to find the values of "b" for which the given system has a nontrivial solution, you need to solve the equations and make sure that you don't end up with a row of all zeroes. To start, you can assign parameters and factor out x1, x2, and x3 to write the system as a matrix. Then, find the determinant of the resulting matrix to determine the values of "b" that will result in a nontrivial solution.
  • #1
alexis36
8
0

Homework Statement


I am asked to find all the values of "b" for which the following system has a nontrivial (i.e non-zero solution).


Homework Equations


bx1 - bx3 = 0
x1 + (b+1)x2 + 2x3 = 0
bx1 + (2b+2) x2 = 0


The Attempt at a Solution


I know that I am needing to solve for the equations so that I dont' end up with a row of all zeroes.
But I am just confused as to where to start. Because I have a bunch of different variables, and I don't know if I should assign parameters to begin with or not?
 
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  • #2
write as a matrix with x1, x2, and x3 factored out then find the determinant of the result.
 
  • #3
thank you!
 

FAQ: LINEAR ALGEBRA: nontrivial solutions

What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their applications using vector spaces and matrices. It involves the analysis of systems of linear equations and their solutions.

What are nontrivial solutions in linear algebra?

Nontrivial solutions are solutions to a system of linear equations that are not simply a repetition of one of the equations or a combination of the other equations. In other words, they are solutions that involve more than just one variable being equal to a constant value.

Why are nontrivial solutions important in linear algebra?

Nontrivial solutions are important because they represent a more complex and interesting solution to a system of linear equations. They allow us to solve for multiple variables and understand the relationships between them. Nontrivial solutions also have practical applications in fields such as physics, engineering, and economics.

How do you find nontrivial solutions in linear algebra?

To find nontrivial solutions, you must first set up a system of linear equations using matrices. Then, you can use techniques such as row reduction and Gaussian elimination to manipulate the equations and solve for the variables. Nontrivial solutions will involve variables that are not all equal to a constant value.

What are some real-world applications of nontrivial solutions in linear algebra?

Nontrivial solutions have many real-world applications, such as in population growth models, network analysis, and image processing. They are also used in computer graphics and machine learning algorithms. Nontrivial solutions allow us to understand complex relationships and make predictions based on data.

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