Finding a Basis for the Kernel Space of a Matrix - Solving the RREF Method

In summary, the basis for the kernel space of the given matrix is { [-2 1 0 0 0]T, [-1 0 3 1 0]T }. This can be found by interpreting the RREF of the matrix as a set of equations and solving for the variables. The vectors in the basis must be orthogonal to each row of the RREF.
  • #1
theRukus
49
0

Homework Statement


Find a basis for the kernel space of the following matrix:
-1 -2 -1 2 2
-2 -4 -4 10 2
1 2 2 -5 2
-1 -2 0 -1 0

row reduce to

1 2 0 1 0
0 0 1 -3 0
0 0 0 0 1
0 0 0 0 0


Somehow read the solution as

{ [-2 1 0 0 0]T, [-1 0 3 1 0]T }

.. I don't understand how to read the basis from the RREF. Could someone shed some light for me? Thanks so much!
 
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  • #2
The basis for the null space is orthogonal to each vector in the rowspace and so to each vector in the rowspace of the rref form
 
  • #3
In more "simple-minded" terms, you can think of your row reduced matrix as referring to
[tex]\begin{bmatrix}1 & 2 & 0 & 1 & 0 \\ 0 & 0 & 1 & -3 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ x_6\end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \\ 0\end{bmatrix}[/tex]

or simply the equations [itex]x_2+ 2x_2+ 0x_3+ x_4+ 0x_5= 0[/itex], [itex]0x_1+ 0x_2+ x_3- 3x_4+ 0x_5= 0[/itex], [itex]0x_1+ 0x_2+ 0x_3+ 0x_4+ x_5= 0[/itex], [itex]0x_1+ 0x_2+ 0x_3+ 0x_4+ 0x_5= 0[/itex].

If you interpret each of those equations as a "dot product" you can see how the vectors in the kernel must be "orthogonal" to the rows of the reduced matrix. Of course, you can also see, from the third equation that [itex]x_5[/itex] must be 0, from the second that [itex]x_3- 3x_4= 0[/itex] so that [itex]x_3= 3x_4[/itex], and from the first equation that [itex]x_1+ 2x_2+ x_4= 0[/itex] so that [itex]x_1= -2x_2- x_4[/itex]. That is, we can write
[tex]\begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5\end{bmatrix}= \begin{bmatrix}-2x_2- x_4 \\ x_2 \\ 3x_4 \\ x_4 \\ 0\end{bmatrix}[/tex]
[tex]= \begin{bmatrix}-2x_2 \\ x_2 \\ 0 \\ 0 \\ 0\end{bmatrix}+ \begin{bmatrix}-x_4 \\ 0 \\ 3x_4 \\ x_4 \\ 0\end{bmatrix}[/tex]
[tex]= x_2\begin{bmatrix}- 2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}+ x_4\begin{bmatrix}-1 \\ 0 \\ 3 \\ 1 \\ 0 \end{bmatrix}[/tex]
 
  • #4
Thanks heaps hallsofivy, that cleared things up for me. Wish me luck on my exam!
 

1. What is the difference between user space and kernel space?

User space and kernel space refer to two different memory spaces within a computer's operating system. User space is where user applications and processes run, while kernel space is a protected area where the operating system and its core functions are stored and executed.

2. Why is the kernel space important?

The kernel space is important because it contains the most critical components of the operating system, including its memory management, process scheduling, and hardware drivers. These components are essential for the functioning of the entire system.

3. What is the role of the kernel in the operating system?

The kernel is the core component of the operating system and acts as a bridge between the hardware and software. It manages all system resources, including memory, CPU, and input/output devices, and ensures that different processes and applications can run smoothly.

4. How does the kernel handle privileged operations?

The kernel has special privileges that allow it to access and control the hardware directly. This allows it to perform essential tasks such as managing memory, scheduling processes, and handling system calls. These privileged operations are tightly controlled to prevent unauthorized access and ensure the security of the system.

5. What is the relationship between the kernel and device drivers?

Device drivers are software components that allow the operating system to communicate with hardware devices. They are loaded into the kernel space and work closely with the kernel to manage hardware resources and handle input/output operations. The kernel relies on device drivers to communicate with external devices and perform tasks on their behalf.

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