# Thread: [SOLVED] Show that the following vectors are two different bases for the following pl

1. ## [SOLVED] Show that the following vectors are two different bases for the following pl

Show that

$\begin{pmatrix} 4\\0\\1 \end{pmatrix}$ , $\begin{pmatrix} 2\\1\\0 \end{pmatrix}$
and
$\begin{pmatrix} 2\\-1\\1 \end{pmatrix}$ , $\begin{pmatrix} 0\\2\\-1 \end{pmatrix}$

are two different bases for the plane

$x-2y-4z=0$

Thanks in advance!

2. Originally Posted by JCS007
Show that

(4 0 1) $.^T$ , (2 1 0) $.^T$ and
(2 -1 1) $.^T$ , (0 2 -1) $.^T$

are two different bases for the plane

$x-2y-4z=0$

Thanks in advance!
Consider an arbitrary vector written in terms of the first basis:
$V = a \left ( \begin{matrix} 4 \\ 0 \\ 1 \end{matrix} \right ) + b \left ( \begin{matrix} 2 \\ 1 \\ 0 \end{matrix} \right )$

Now show that $V_x - 2V_y - 4V_z = 0$ for all a and b.

The proof for the other basis is similar.

-Dan