How are the 2 vectors linearly dependent or linearly independent?
The question is
Examine whether the vector i-j+2k,2i+j-3k, 2i-5j+k are linearly independent or linearly dependent?
You need compose a determinant from the coordinates of these vectors.Examine whether the vector i-j+2k,2i+j-3k, 2i-5j+k are linearly independent or linearly dependent?
If a determinant is equal to 0, then vectors are linearly dependent, and conversely, vectors are linearly independent if a determinant is not equal to 0.
For your problem $\displaystyle i-j+2k, 2i+j-3k, 2i-5j+k$ it must be $\displaystyle \left| {\begin{array}{*{20}{c}}1 & { - 1} & 2 \\2 & 1 & { - 3} \\2 & { - 5} & 1 \\\end{array} } \right|$.