Solving a linear set of unknown Vectors

In summary, the person is asking for a mathematical procedure to solve for two vectors in a set of linear vector equations. They mention a general equation for solving linear equations and ask for clarification on the meaning of Ax. They also express concern about the underdetermined nature of the system.
  • #1
m26k9
9
0
Hello,

I am trying to find a mathematical procedure for finding the solutions for a linear set of vector equations.

For example I have;

Ax + By = P (1)
Cx + Dy = Q (2)

Here, A,B,C,D,x and y are all Nx1 vectors. So I need to solve for two Nx1 vectors.
For a general linear equations like Ax=B -> x=A^-1.B, is there any standard procedure to solve for vectors of unknowns?

Thank you very much.
 
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  • #2
If A and x are column (?) vectors, then what is Ax? Is it the inner product?
Because in that case, you have two equations for 2N unknowns which will give you a heavily underdetermined system.
 

1. What is a linear set of unknown vectors?

A linear set of unknown vectors is a collection of vectors in which the coefficients of the variables are unknown. These types of systems can be represented by a system of linear equations and can be solved using various methods such as substitution, elimination, or matrix operations.

2. How do you solve a linear set of unknown vectors using substitution?

To solve a linear set of unknown vectors using substitution, you must first solve for one variable in one of the equations. Then, substitute that value into the other equations, simplifying the system to a single variable. Continue this process until all variables have been solved for.

3. What is meant by "consistent" and "inconsistent" in a linear set of unknown vectors?

A consistent system is one in which there is at least one solution that satisfies all of the equations. An inconsistent system is one in which there is no solution that satisfies all of the equations. This can occur when the equations are contradictory or when there are not enough equations to solve for all of the variables.

4. How do matrix operations help in solving a linear set of unknown vectors?

Matrix operations, such as Gaussian elimination, provide a systematic way of solving a system of linear equations. By converting the equations into a matrix, we can use row operations to simplify the system and solve for the unknown variables. This method is especially useful for larger systems with many equations and variables.

5. Can a linear set of unknown vectors have more than one solution?

Yes, a linear set of unknown vectors can have infinite solutions or no solutions at all. This depends on the number of equations and variables in the system. If there are more variables than equations, there will be infinite solutions. If there are more equations than variables, there may be no solutions. If the number of equations and variables are equal, there will either be a unique solution or no solution.

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