Linear algebra proof, dependent or independent Suppose that $p_1=4-3x+6x^2+2x^3$, $p_2=1+8x+3x^2+x^3$, and $p_3=3-2x-x^2$ are vectors in $P_3$. Determine if $p_1$,$p_2$,and $p_3$ are linearly independent or dependent. Justify your answer.
So far, what I did was say:
Suppose $k_1$, $k_2$ and $k_3$ are constants.
Then I said:
$$k_1 
\begin{pmatrix}
4\\ 
-3\\ 
2\\ 
6
\end{pmatrix}
+ k_2 
\begin{pmatrix}
1\\ 
8\\ 
3\\ 
1
\end{pmatrix}
+ k_3 \begin{pmatrix}
3\\ 
-2\\ 
-1\\ 
0
\end{pmatrix}$$
Then I made the augmented matrix $$\begin{pmatrix}
4 & 1 & 3 &0 \\ 
-3 & 8 & -2 & 0\\ 
6 & 3 & -1 & 0\\ 
2 & 1 & 0 & 0
\end{pmatrix}$$
and found that the RREF is 
\begin{pmatrix}
1 & 0 & 0 & 0\\ 
0 & 1 & 0 & 0\\ 
0 & 0 & 1 & 0\\ 
0 & 0 & 0 & 0
\end{pmatrix}
Therefore, $k_1 = k_2 = k_3 = 0$.
Therefore, $p_1$, $p_2$ and $p_3$ are linearly independent. 
So far this is what I have. If this is right then how to I go about justifying my answer?
Also, is there a better way of writing vectors on this website? 

That absolutely justifies that $p_1,p_2,p_3$ are linearly independent.
