[Math] Basis for the subspace w spanned by ${v_1,v_2,v_3,v_4}$.

Question

My textbook doesn't contain any solution to the answer so I was wondering if my answer is right.

Let $$v_1 =\begin{bmatrix}
1\\
-3 \\
4
\end{bmatrix},
v_2 = \begin{bmatrix}
6 &\\
2\\
-1
\end{bmatrix},
v_3 = \begin{bmatrix}
2 &\\
-2\\
3
\end{bmatrix},
\text { and } v_4 = \begin{bmatrix}
-4 &\\
-8\\
9
\end{bmatrix}$$

Find a basis for the subspace $W$ spanned by ${v_1,v_2,v_3,v_4}$?

What I did is that I reduced the matrix:

$$W = \begin{bmatrix}
1 & 6 & 2 &-4\\
-3 & 2 &-2 &-8\\
4 &-1 & 3 & 9
\end{bmatrix}$$

then I reduced it rref –> $$W = \begin{bmatrix}
1 & 0 & 8 &-2\\
0 & 1 & \frac{4}{20} &-1\\
0 & 0 & 0 & 0
\end{bmatrix}$$

therefore the basis for the subspace $W$ spanned by ${v_1,v_2,v_3,v_4}$. would be $B= {\begin{bmatrix}
1 \\
-3 \\
4
\end{bmatrix},\begin{bmatrix}
6 \\
2 \\
-1
\end{bmatrix} }$

Is this correct?

Answer 1

This question was answered in comments:

Yes, it seems correct. You can verify that indeed $v_3,v_4\in \text{span}(v_1,v_2)$. – Berci Jan 27 at 21:11

and

Yes it is correct. Minor mistakes in your row reduction but doesn't affect the final results. See here for full row reduction. – John Habert Jan 27 at 21:16