1
$\begingroup$

I am working on a HW assignment for a linear algebra course and I want to make sure that the way I am determining the kernel of a matrix and presenting the results is correct. Here is the problem and my solution:

For the following matrix, write the kernel as the span of a finite number or vectors. Is the kernel a point, line, plane or all of $\mathbb{R}^{3}$

$$ \begin{bmatrix} 2 & 6 & -4 \\ -1 & -3 & 2 \end{bmatrix} $$

The first step is to put the matrix into row-echelon form

$$\begin{bmatrix} 2 & 6 & -4 \\ -1 & -3 & 2 \end{bmatrix} \quad \xrightarrow{R_{1} + 2 R_{2}} \quad \begin{bmatrix} 2 & 6 & -4 \\ 0 & 0 & 0 \end{bmatrix} $$

$x_{2}$ and $x_{3}$ are now free, so we need to write an equation for $x_{1}$ in terms of $x_{2}$ and $x_{3}$

$$ 2x_{1} + 6x_{2} - 4x_{3} = 0 \quad\to\quad 2x_{1} = -6x_{2} + 4_{x3} \quad\to\quad x_{1} = -3x_{2} + 2x_{3} $$

this now allows us to determine the kernel

$$ \vec{x} \,\,=\,\, \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} \,\,=\,\, \begin{bmatrix} -3x_{2} + 2x_{3} \\ x_{2} \\ x_{3} \end{bmatrix} \,\,=\,\, x_{2} \begin{bmatrix} -3 \\ 1 \\ 0 \end{bmatrix} \,\,+ \,\, x_{3} \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} $$

the kernel is $\begin{bmatrix} -3 & 1 & 0 \end{bmatrix}^{T} ,\, \begin{bmatrix} 2 & 0 & 1 \end{bmatrix}^{T}$ and is a plane.

As mentioned before, this is a HW assignment so I am simply looking for advice and, if needed, guidance in the right direction.

Thank you in advance for any help

$\endgroup$
  • 1
    $\begingroup$ Maybe say that the kernel is spanned by those two vectors, and it looks good, very good! $\endgroup$ – pjs36 Feb 25 '15 at 2:55
  • $\begingroup$ Okay, I already knew the answer was correct (it's in the back of the book) I just wanted to make sure that the way I was finding the kernel and presenting the results was adequate enough to receive full credit. $\endgroup$ – Linear Algebra Student Feb 25 '15 at 3:03
-1
$\begingroup$

That's certainly correct, you need nothing more!

$\endgroup$
  • $\begingroup$ Thank you, I just hate losing points for trivial errors. $\endgroup$ – Linear Algebra Student Feb 25 '15 at 3:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.