Convert infinite solution to vector form

In summary, the solution for the given system of equations has an infinite number of solutions, with $x_3$ and $x_4$ being the free variables. The general solution can be written in vector form as $x = \begin{bmatrix}4/3\\2\\0\\0\end{bmatrix} + x_3\begin{bmatrix}1/3\\1/3\\1\\0\end{bmatrix} + x_4\begin{bmatrix}-5/3\\1/3\\0\\1\end{bmatrix}$, where $x_3$ and $x_4$ can take on any value. Alternatively, the free variables can be
  • #1
ibwev
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0
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I know the solution has an infinite number of solutions. It is represented as follows:

x1= 4/3 + (1/3)x3 - (5/3)x4
x2= 2 + (1/3)x3 + (1/3)x4
x3= Free
x4= Free

How do I put the above solution into vector form as illustrated in the original question?
 

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  • #2
Hi ibwev,

If you set the free variables $x_3, x_4$ equal to zero, you find that $\begin{bmatrix}4/3 & 2 & 0 & 0\end{bmatrix}^T$ is a particular solution. So the general solution can be written$$x = \begin{bmatrix}4/3\\2\\0\\0\end{bmatrix} + x_3\begin{bmatrix}1/3\\1/3\\1\\0\end{bmatrix} + x_4\begin{bmatrix}-5/3\\1/3\\0\\1\end{bmatrix}.$$
 
  • #3
Thank you so much. I have been trying to figure this out for 2 days. I really appreciate the help.
 
  • #4

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  • #5
The $1$'s come from the trivial equations $x_3 = x_3$ and $x_4 = x_4$.
 
  • #6
Euge said:
The $1$'s come from the trivial equations $x_3 = x_3$ and $x_4 = x_4$.

Most people use a parameter for the free variables, such as $ t = x_4 $ and $ s = x_3 $. It is much easier to see the vector form of the line of solution (or in this case, hyper-line of solution).
 
  • #7
I want to make it clear that in practice I use different letters in place of the free variables to express the general solution parametrically.
 

FAQ: Convert infinite solution to vector form

What does it mean to convert an infinite solution to vector form?

Converting an infinite solution to vector form involves representing the solution to a system of equations as a vector, which is a mathematical object that contains both magnitude and direction. This allows for a more concise and organized way to describe the solution.

Why is it important to convert an infinite solution to vector form?

Converting an infinite solution to vector form can provide a more efficient and clear representation of the solution. It also allows for easier manipulation and calculation of the solution.

What is the process for converting an infinite solution to vector form?

The process for converting an infinite solution to vector form involves identifying the variables and coefficients in the system of equations, and then organizing them into a vector using matrix operations. This typically involves using Gaussian elimination or other methods of solving systems of equations.

What are the benefits of representing a solution as a vector?

Representing a solution as a vector can make it easier to visualize and understand. It also allows for easier comparison and analysis of different solutions, as well as the ability to use vector operations to manipulate the solution.

Are there any limitations to converting an infinite solution to vector form?

While converting an infinite solution to vector form can be beneficial, it may not always be possible for more complex systems of equations. Additionally, the process may become more complicated and time-consuming for larger systems.

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