Consider the vectors \begin{bmatrix} 1 \\ 2 \\ 7 \end{bmatrix}, \begin{bmatrix} 3 \\ -1 \\ 8 \end{bmatrix}, \begin{bmatrix} 0 \\ 5 \\ 4 \end{bmatrix}, \begin{bmatrix} 7 \\ 20 \\ 77 \end{bmatrix}. Are the vectors linearly dependent or independent? Hint: Note that we have 4 vectors in \mathbb{R}^3 and therefore the system x_1 \begin{bmatrix} 1 \\ 2 \\ 7 \end{bmatrix}+ x_2 \begin{bmatrix} 3 \\ -1 \\ 8 \end{bmatrix}+x_3 \begin{bmatrix} 0 \\ 5 \\ 4 \end{bmatrix}+ x_4 \begin{bmatrix} 7 \\ 20 \\ 77 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} is homogeneous and underdetermined (read carefully the NOTE in the slide)