Answer
The vectors are linearly dependent.
Work Step by Step
We use theorem 4 and the invertibility theorem, that together state, that a set of vectors are only linearly independent if and only if the matrix they form have a determinant different from $0$.
We use the trick in the picture:
$$ \left|\begin{matrix} 4&-7&-3\\6&0&-5\\2&7&-2 \end{matrix}\right|=\\4\cdot0\cdot(-2)+(-7)\cdot(-5)\cdot2+(-3)\cdot6\cdot7-(-3)\cdot0\cdot2-7\cdot(-5)\cdot4-(-2)\cdot6\cdot(-7)=\\0+70-126+0+140-84=0$$
Since the determinant is equal to $0$, then we have, from theorem 4, that the matrix is not invertible, and therefore the vectors are linearly dependent.
