Linear Algebra Homogeneous Linear Equations

[CoachMorty]

Hello everyone, this is my first post to the forum so I apologize in advance if I have placed this topic in the wrong section. I'm looking for some help with homogenous equations, I just can't seem to understand how to work even the simplest of problems. I have posted the problems below, I would appreciate if one or all 3 could be worked out step-by-step to help me understand the best I can. Thanks for your time and I appreciate the help in advance

homogeneous system In Exercises 43–46, solve the homogeneous linear system corresponding to the given coefficient matrix.

43. [100]
-----[011]
-----[000]

44. [1000]
-----[0110]

45. [1001]
-----[0010]
-----[0000]

Solutions from the book

43.
x1= 0
x2= -t
x3= t

44. odds only

45.
x1= -t
x2= s
x3= 0
x4= t

 

[Deleted member 4993]

Hello everyone, this is my first post to the forum so I apologize in advance if I have placed this topic in the wrong section. I'm looking for some help with homogenous equations, I just can't seem to understand how to work even the simplest of problems. I have posted the problems below, I would appreciate if one or all 3 could be worked out step-by-step to help me understand the best I can. Thanks for your time and I appreciate the help in advance

homogeneous system In Exercises 43–46, solve the homogeneous linear system corresponding to the given coefficient matrix.

43. [100]
-----[011]
-----[000]

44. [1000]
-----[0110]

45. [1001]
-----[0010]
-----[0000]

Solutions from the book

43.
x1= 0
x2= -t
x3= t

44. odds only

45.
x1= -t
x2= s
x3= 0
x4= t

Can you please tell us the definition of homogeneous system?

What are the mathematical properties attributed to such system?

 

[Steven G]

Hello everyone, this is my first post to the forum so I apologize in advance if I have placed this topic in the wrong section. I'm looking for some help with homogenous equations, I just can't seem to understand how to work even the simplest of problems. I have posted the problems below, I would appreciate if one or all 3 could be worked out step-by-step to help me understand the best I can. Thanks for your time and I appreciate the help in advance

homogeneous system In Exercises 43–46, solve the homogeneous linear system corresponding to the given coefficient matrix.

43. [100]
-----[011]
-----[000]

44. [1000]
-----[0110]

45. [1001]
-----[0010]
-----[0000]

Solutions from the book

43.
x1= 0
x2= -t
x3= t

44. odds only

45.
x1= -t
x2= s
x3= 0
x4= t

#43: Assuming the variables from left to right are x₁, x₂ and x₃, can you please tell us the equation that row 1 gives us? How about row 2 and row 3?

 

[mmm4444bot]

I don't know from where symbols s and t come. :???:

@CoachMorty: Please read the forum's guidelines (here is a summary) and make sure you've posted all of the given information. Thank you. :cool:

 

[Steven G]

I don't know from where symbols s and t come. :???:

Hmm, I am sure that you know that x₂ and x₄ can be any real numbers such as s and t repectively

 

[mmm4444bot]

Hmm, I am sure that you know that x₂ and x₄ can be any real numbers such as s and t repectively

Oops. (Didn't study the exercises.) Are symbols 's' and 't' standard for this use, in linear algebra? Maybe I'm too accustomed to using letters near the beginning of the alphabet for constants.

 

[Steven G]

Oops. (Didn't study the exercises.) Are symbols 's' and 't' standard for this use, in linear algebra? Maybe I'm too accustomed to using letters near the beginning of the alphabet for constants.

Yes, these are standard letters.

 

[HallsofIvy]

Hello everyone, this is my first post to the forum so I apologize in advance if I have placed this topic in the wrong section. I'm looking for some help with homogenous equations, I just can't seem to understand how to work even the simplest of problems. I have posted the problems below, I would appreciate if one or all 3 could be worked out step-by-step to help me understand the best I can. Thanks for your time and I appreciate the help in advance

Do you know what "homogeneous equations" and "homogeneous systems" are? Do you know how a "coefficient matrix" corresponds to a homogeneous system?

homogeneous system In Exercises 43–46, solve the homogeneous linear system corresponding to the given coefficient matrix.

43. [100]
-----[011]
-----[000][COLOR=#3c3c3c[/FONT]

Writing the variables as \(\displaystyle x_1,\), \(\displaystyle x_2\), and \(\displaystyle x_3\) that coefficient matrix corresponds to the "homogeneous matrix equation \(\displaystyle \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix}\begin {bmatrix}x_1 \\ x_2\\ x_3 \end{bmatrix}= \begin{bmatrix}x_1 \\ x_2+ x_3 \\ 0 \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\) which leads to the equations \(\displaystyle x_1= 0\), \(\displaystyle x_2+ x_3= 0\) and 0= 0. From \(\displaystyle x_2+ x_3= 0\) we have \(\displaystyle x_2= -x_1\).

44. [1000]
[/COLOR]

-----[0110]

This matrix corresponds to the equations \(\displaystyle x_1= 0\) and \(\displaystyle x_2+ x3= 0\). Since there are 4 columns in the matrix there 4 unknowns but we are given no information about \(\displaystyle x_4\) so it could be any thing.

45. [1001]
-----[0010]
-----[0000]

This matrix corresponds to the equations \(\displaystyle x_1+ x_4= 0\), which is the same as \(\displaystyle x_1= -x_4\), \(\displaystyle x_3= 0\) and 0= 0. That last equation gives us no information. Do you understand how I got the equations from the "coefficient matrix"?

Solutions from the book

43.
x1= 0

x2= -t
x3= t

As I said above, the "coefficient matrix" gives the equations \(\displaystyle x_1= 0\) and \(\displaystyle x_2= -x_3\). The first of those equations gives the "x_1= 0" above. The second says "whatever \(\displaystyle x_3\), \(\displaystyle x_2\) is the negative. But \(\displaystyle x_3\) can be anything. The person who wrote this solution chose to call the value of \(\displaystyle x_3\) "t" which can be any number and then \(\displaystyle x_2= -t\).

44. odds only

The equations, above, were \(\displaystyle x_1= 0\) and \(\displaystyle x_2+ x_4= 0\). Those are exactly the same as in problem 43 so we can write the same solution, \(\displaystyle x_1= 0\), \(\displaystyle x_2= -t\), and \(\displaystyle x_3= t\) where t can be any number. The fact that there is a fourth column implies there is fourth unknown, \(\displaystyle x_4\) but since it is not in any equation, \(\displaystyle x_4\) can be any number. We might write "\(\displaystyle x_4= s\)" where "s" just represents some unknown number. We don't use "t" again because we don't want to imply that \(\displaystyle x_4\) must be the same as \(\displaystyle x_3\).

45.
x1= -t
x2= s
x3= 0
x4= t

The matrix corresponds, as above, to the equations \(\displaystyle x_1= -x_4\) and \(\displaystyle x_3= 0\). There is an "\(\displaystyle x_2\)" of course but it doesn't appear in any equation so could be anything. That is why we have "\(\displaystyle x_2= s\). It can be any number. And, as before, "\(\displaystyle x_1= -x_4\)" is represented by taking "\(\displaystyle x_4= t\)", again representing any number, and "\(\displaystyle x= -t\)".