<s>[INST] Finding basis for the space spanned by some vectors. Find a subset of vectors $\{v_1, v_2, v_3, v_4, v_5\}$ that forms the basis for the space spanned by these vectors: $$v1=\left ( \begin{array}{c} 1\\-2\\0\\3 \end{array}\right), v2= \left ( \begin{array}{c} 2\\-5\\-3\\6\end{array}\right)\, ,\, v3=\left ( \begin{array}{c}1\\-1\\3\\1\end{array} \right)\, ,\, v4=\left(\begin{array}{c} 2\\-1\\4\\-7\end{array} \right )\, ,\, v5=\left (\begin{array}{c}3\\2\\14\\-17\end{array} \right).$$
 [/INST] You don't need to guess; just write down the matrix having the vectors as columns:
$$\begin{bmatrix}
1 & 2 & 1 & 2 & 3 \\
-2 & -5 & -1 & -1 & 2 \\
0 & -3 & 3 & 4 &  14  \\
3 & 6 & 1 & -7 &  -17
\end{bmatrix}$$
and proceed with Gaussian elimination; first do $R_2+2R_1$ (sum to the second row the first multiplied by $2$) and then $R_4+(-3)R_1$ to get
$$\begin{bmatrix}
1 & 2 & 1 & 2 & 3 \\
0 & -1 & 1 & 3 & 8 \\
0 & -3 & 3 & 4 &  14  \\
0 & 0 & -2 & -13 &  -24
\end{bmatrix}$$
I usually do pivot reduction, so multiply the second row by $-1$ and then do $R_3+3R_2$ to get
$$\begin{bmatrix}
1 & 2 & 1 & 2 & 3 \\
0 & 1 & -1 & -3 & -8 \\
0 & 0 & 0 & -5 &  -10  \\
0 & 0 & -2 & -13 &  -24
\end{bmatrix}$$
Now swap the third and fourth rows; if you also do pivot reduction you get
$$\begin{bmatrix}
1 & 2 & 1 & 2 & 3 \\
0 & 1 & -1 & -3 & -8 \\
0 & 0 & 1 & 13/2 &  12 \\
0 & 0 & 0 & 1 &  2  \\
\end{bmatrix}$$
Since we have pivots in the first four columns, we conclude that $v_1, v_2, v_3, v_4$ span your subspace. But, of course, since the dimension of the subspace is $4$, it is the whole $\mathbb{R}^4$, so any basis of the space would do.
These computations are surely easier than computing the determinant of a $4\times 4$ matrix. Note that if the dimension of the subspace were less than $4$, computing a determinant built with any set of four vectors would lead to nothing, while the elimination always works.
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