Find the kernel of a 4x4 matrix $$
    \begin{pmatrix}
    1 & 2 & 3 & 4\\
    5 & 6 & 7 & 8\\
    9 & 10 & 11 & 12\\
    13 & 14 & 15 & 16\\
    \end{pmatrix}
$$
I am asked to find the kernel of the matrix $M$. After doing some row operation I get to 
$$
    \begin{pmatrix}
    1 & 2 & 3 & 4\\
    0 & -4 & -8 & -12\\
    0 & 0 & 0 & 0\\
    0 & 0 & 0 & 0\\
    \end{pmatrix}
$$
and for $x$ I find $x = \alpha + 2\beta$, whereas $y = -2\alpha -3\beta$
Therefore, 
$$
    \begin{pmatrix}
    \alpha + 2\beta\\
    -2\alpha - 3\beta\\
    \alpha\\
    \beta\\
    \end{pmatrix}
$$
When we take outside alpha and beta:
we get two vectors:
$$
    \begin{pmatrix}
    1\\
    -2\\
    1\\
    0\\
    \end{pmatrix}
$$
and
$$
    \begin{pmatrix}
    2\\
    -3\\
    0\\
    1\\
    \end{pmatrix}
$$
which are linearly independent and form a basis of this $ker(M)$
Could you please confirm with me whether you get the same result? Thank you.

Your computations are correct and so is your answer. But you should really improve your MathJax skills.
