how to determine the solution space of a linear system?

[msegling]

I have completed the question with explanations, however i would like confirm that my thought process is correct.
Please supply any corrections that i could make.


Question


Determine the solution space, described as the span of a set of vectors, for the following linear system:


1x + 5y + 0z = 0
6x + 7y + 9z = 0
0x + 5y + 5z = 0


State, with justification, the dimension of the solution space.




My Attempt:


Row echelon form of the augmented matrix:


| 1 4 0 0 |
| 0 1 19/3 0 |
| 0 0 1 0 |


therefore :
x = -4y;
y = -19/3z;
z = 0;


Let A = the solution set of the augmented matrix.


A= {(x,y,z) | x,y,z ∈ R^3}
= {(-4y,-19/3z, 0) | x,y,z ∈ R^3}
= {(-4y, 0, 0) + (0, -19/3z, 0) | x,y,z ∈ R^3}
= {y(-4, 0, 0) + z(0, -19/3, 0) | x,y,z ∈ R^3}
= span{(-4, 0, 0), (0, -19/3, 0)}


the solution space is thus a subspace on R^3




I'm second guessing my self on the validity of this, any suggestions will be gladly accepted.

 

[Ishuda]

I have completed the question with explanations, however i would like confirm that my thought process is correct.
Please supply any corrections that i could make.


Question


Determine the solution space, described as the span of a set of vectors, for the following linear system:


1x + 5y + 0z = 0
6x + 7y + 9z = 0
0x + 5y + 5z = 0


State, with justification, the dimension of the solution space.




My Attempt:


Row echelon form of the augmented matrix:


| 1 4 0 0 |
| 0 1 19/3 0 |
| 0 0 1 0 |
...

Assume that is true [it is not what I get]. Now if you get z=0 then the first two rows give
\(\displaystyle \begin{pmatrix}{x\, + 4\, y} \\ y \end{pmatrix}\, =\, \begin{pmatrix}0\\0\end{pmatrix}\)
which gives y=0 and the first row then gives
x = 0

 

[msegling]

Assume that is true [it is not what I get]. Now if you get z=0 then the first two rows give
\(\displaystyle \begin{pmatrix}{x\, + 4\, y} \\ y \end{pmatrix}\, =\, \begin{pmatrix}0\\0\end{pmatrix}\)
which gives y=0 and the first row then gives
x = 0

Thank you for your feedback, do you mind attempting the question?

 

[Ishuda]

Thank you for your feedback, do you mind attempting the question?

Given
1x + 5y + 0z = 0
6x + 7y + 9z = 0
0x + 5y + 5z = 0

To reduce to the matrix start with subtracting 6 times the first row from the second to get
1x + 5y + 0z = 0
0x -23y + 9z = 0
0x + 5y + 5z = 0

Now multiply the third row by 23/5 and add the second row to it to get
1x + 5y + 0z = 0
0x -23y + 9z = 0
0x + 0y +32z = 0

Multiply the second row by -1/23 and divide the third row by 32 to get
1x + 5y + 0z = 0
0x + 1y -(9/23)z = 0
0x + 0y + 1z = 0

So z=0, ...

BTW: A point has zero dimension.

 

[msegling]

Given
1x + 5y + 0z = 0
6x + 7y + 9z = 0
0x + 5y + 5z = 0

To reduce to the matrix start with subtracting 6 times the first row from the second to get
1x + 5y + 0z = 0
0x -23y + 9z = 0
0x + 5y + 5z = 0

Now multiply the third row by 23/5 and add the second row to it to get
1x + 5y + 0z = 0
0x -23y + 9z = 0
0x + 0y +32z = 0

Multiply the second row by -1/23 and divide the third row by 32 to get
1x + 5y + 0z = 0
0x + 1y -(9/23)z = 0
0x + 0y + 1z = 0

So z=0, ...

BTW: A point has zero dimension.

right, so how would i go about determining the span ?

 

[Ishuda]

right, so how would i go about determining the span ?

What is the span of a single unique solution to the problem?