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Examine whether the vectors X1 = [3 1 1], X2 = [2 0 -1], X3 = [4 2 1] are linearly independent.
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Matrix Equation

k1[3,1,1]+k2[2,0,1]+k3[4,2,1]=03k1+2k2+4k3=01k1+0k2+2k3=0k1k2+k3=0 [324 102\111][k1k2\k3]=[00\0]   R_{12}\\begin{bmatrix} 1&0&2 \ 3&2&4 \1&-1&1 \end{bmatrix} \begin{bmatrix}k_1\\ k_2\k_3\end{bmatrix}= \begin{bmatrix}0\\ 0\0\end{bmatrix} R23R1,R3R1[102022011][k1 k2k3]=[0 00]

$R_2+2R_1,\\\begin{bmatrix} 1&0&2 \\ 0&0&-2\\0&-1&-1 \end{bmatrix} \begin{bmatrix}k_1\\\ k_2\\k_3\end{bmatrix}= \begin{bmatrix}0\\\ 0\\0\end{bmatrix} \\ k_1 +2k_3=0\\ …

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