Solving a system of linear equations using back substitution

In summary, the term "target vector" is commonly used in linear algebra to refer to the desired solution of a system of linear equations. This can be in cases where there is no feasible solution and a least squared error solution is desired, or when finding a single vector of inputs that gives the best least-squared-error solution to multiple target vectors at the same time. This term can also be used when there is an exact solution with no errors, and the back-substitution process is used to find a single solution with a single "target" vector. Different error metrics can also be used in this context.
  • #1
Jen2114
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Hello,
its been a while since I have taken linear algebra and I am having trouble understanding what a target vector is. I need to solve a system of linear equations in matrix form using back substitution and with different target vectors. I don't have a problem with back substitution, but I don't recall exactly what a target vector is and how I would use it with back substitution to solve the system. Could someone help me clarify this concept? Thanks!
 
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  • #2
The only use of the term "target vector" that I have seen is where there is no feasible solution of the linear equations and a least squared error solution is desired.

But I imagine that there is no reason not to use that term also when there is an exact solution with no errors. In that case, a single solution with a single "target" vector would be the back-substitution process that you are familiar with.

Another possibility is that you are being asked to find a single vector of inputs that gives the best least-squared-error solution to multiple target vectors at the same time.

PS. The fact that I mention lease-squared-error is simply because that is the most common error minimization problem and a lot of effort has been exerted to find quadratic minimization algorithms to solve that problem. Other error metrics can also be used.
 

What is back substitution?

Back substitution is a method used to solve a system of linear equations by starting from the last equation and solving for one variable at a time, working backwards through the system until all variables have been solved for.

When is back substitution used?

Back substitution is typically used when the system of linear equations is triangular, meaning all of the equations are in the form of x = a or y = a + bx. It is also used when the system has been reduced using elimination or other methods.

What are the steps for solving a system of linear equations using back substitution?

The steps for solving a system of linear equations using back substitution are as follows:

  1. Start with the last equation in the system.
  2. Solve for the variable in that equation.
  3. Substitute the value found in step 2 into the second to last equation.
  4. Solve for the variable in the second to last equation.
  5. Repeat steps 3 and 4 until all variables have been solved for.

What are the advantages of using back substitution?

One advantage of using back substitution is that it is a relatively simple and straightforward method for solving systems of linear equations, especially when the system is already in triangular form. It also allows for the equations to be solved one at a time, making it easier to keep track of the steps and avoid errors.

Are there any limitations to using back substitution?

Back substitution can only be used on systems of linear equations that are triangular or have been reduced using other methods. It also may not be the most efficient method for solving larger systems, as it requires working backwards through the equations. Additionally, back substitution may not always produce a unique solution for the system.

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