Determining Which Variables Are Free

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In summary, the matrix from problem 39 can be deleted without affecting the remaining matrix columns, which span \mathbb{R}^4.
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Homework Statement


Find the column of the matrix in Exercise 39 that can be deleted and yet have the remaining matrix columns span [itex]\mathbb{R}^4[/itex]

Homework Equations


The matrix from problem 39 is

[itex]\begin{bmatrix}
10 & -7 & 1 & 4 & 6 \\
-8 & 4 & -6 & -10 & -3 \\
-7 & 11 & -5 & -1 & -8 \\
3 & -1 & 10 & 12 & 12 \\
\end{bmatrix}

[/itex]

The Attempt at a Solution



I had MATLAB compute the row-reduced echelon form of the matrix augmented with the arbitrary point/vector [itex]\mathbf{b}[/itex], where [itex]\mathbf{b} \in \mathbb{R}^4[/itex]. The result is given as an attachment.

I was wondering, does this augmented matrix contain any free variables. I believe it does, and allow me to explain:

The augmented matrix can be written as

[itex]\begin{bmatrix}
1 & 0 & 0 & 1 & 0 & f_1(b) \\
0 & 1 & 0 & 1 & 0 & f_2(b) \\
0 & 0 & 1 & 1 & 0 & f_3(b) \\
0 & 0 & 0 & 0 & 1 & f_4(b) \\
\end{bmatrix}[/itex]

Writing out the system of linear equations, that this augmented matrix represents, gives

[itex]\begin{array} \\
x_1 & + & x_2 & + & x_3 & + & x_4 & = & f_1(b) \\
& & x_2 & + & x_3 & & & = & f_2(b) \\
& & & & x_3 & + & x_4 & = & f_3(b) \\
& & & & & & x_4 & = & f_4(b) \\

\end{array}[/itex]

If I understand what a free variable is, it is one of which can be written in terms of the others. In this example, we have the other variables can be written as a function of the variables x_4. So, we the variable x_4 is free because you can choose any value of it, and find the corresponding b_i's will be; then from these b_i's, x_3, x_2, and x_1 can be determined.

Am I wrong?
 

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  • #2
Bashyboy said:

Homework Statement


Find the column of the matrix in Exercise 39 that can be deleted and yet have the remaining matrix columns span [itex]\mathbb{R}^4[/itex]


Homework Equations


The matrix from problem 39 is

[itex]\begin{bmatrix}
10 & -7 & 1 & 4 & 6 \\
-8 & 4 & -6 & -10 & -3 \\
-7 & 11 & -5 & -1 & -8 \\
3 & -1 & 10 & 12 & 12 \\
\end{bmatrix}

[/itex]



The Attempt at a Solution



I had MATLAB compute the row-reduced echelon form of the matrix augmented with the arbitrary point/vector [itex]\mathbf{b}[/itex], where [itex]\mathbf{b} \in \mathbb{R}^4[/itex]. The result is given as an attachment.

I was wondering, does this augmented matrix contain any free variables. I believe it does, and allow me to explain:

The augmented matrix can be written as

[itex]\begin{bmatrix}
1 & 0 & 0 & 1 & 0 & f_1(b) \\
0 & 1 & 0 & 1 & 0 & f_2(b) \\
0 & 0 & 1 & 1 & 0 & f_3(b) \\
0 & 0 & 0 & 0 & 1 & f_4(b) \\
\end{bmatrix}[/itex]

Writing out the system of linear equations, that this augmented matrix represents, gives

[itex]\begin{array}
x_1 & + & x_2 & + & x_3 & + & x_4 & = & f_1(b) \\
& & x_2 & + & x_3 & & & = & f_2(b) \\
& & & & x_3 & + & x_4 & = & f_3(b) \\
& & & & & & x_4 & = & f_4(b) \\

\end{array}/itex]

If I understand what a free variable is, it is one of which can be written in terms of the others. In this example, we have the other variables can be written as a function of the variables x_4. So, we the variable x_4 is free because you can choose any value of it, and find the corresponding b_i's will be; then from these b_i's, x_3, x_2, and x_1 can be determined.

Am I wrong?

I your augmented form is right then your answer is also right.
 
  • #3
So, then x_4 is the only free variable?
 
  • #4
Bashyboy said:
So, then x_4 is the only free variable?

It is the free variable for the particular row and column ordering you have selected. If you permute the columns you might come up with a different free variable.

I don't think looking at free variables is the way to go in this question. You want to know which four columns of the matrix form a basis, so the question you need to answer is this: if you omit one of the columns, do you have an invertible matrix? Your final echelon form has the identity matrix in columns 1,2,3,5, so if you leave out column 4 you get the identity matrix. That means that columns 1,2,3,5 form a non-singular matrix.
 
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  • #5
No, I realize that determining which variables are free won't aid in my solving the actual problem. I was simply curious. Thank you.
 

Related to Determining Which Variables Are Free

1. How do you determine which variables are free in a scientific study?

To determine which variables are free in a scientific study, researchers typically use statistical analysis techniques such as regression or correlation. These methods help identify relationships between different variables and determine which ones have a significant impact on the outcome of the study.

2. Why is it important to identify free variables in a study?

Identifying free variables in a study is important because it helps researchers understand which factors are influencing the outcome of the study. By controlling for these variables, researchers can ensure that their results are accurate and not affected by other factors.

3. What are some common examples of free variables in scientific research?

Common examples of free variables in scientific research include age, gender, socioeconomic status, and environmental factors. These variables are often not under the control of the researcher and can have a significant impact on the outcome of the study.

4. Can free variables change during the course of a study?

Yes, free variables can change during the course of a study. This is one of the reasons why it is important to identify and control for these variables. Changes in free variables can affect the results of a study and make it difficult to draw accurate conclusions.

5. How do you control for free variables in a study?

Researchers can control for free variables in a study by using various methods such as randomization, matching, or statistical techniques like analysis of covariance. These methods help reduce the impact of free variables on the outcome of the study and improve the accuracy of the results.

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