I'm having a bit of trouble with this problem. I know what linear dependence implies, but I'm not exactly sure how to apply it to this problem to investigate the second set given to us.
Suppose that S = {v1, v2, v3} is a linearly independent set of three vectors from C^347. Is the set T = {2v1 + v2 − v3, 2v1 + 3v2 + v3, v1 − v2 + 2v3} linearly dependent or linearly independent?
If $\mathbf S$ is the set=matrix of independent column vectors and $$ \mathbf{T} = \left( {\begin{array}{*{20}c} 2 & 1 & { - 1} \\ 2 & 3 & 1 \\ 1 & { - 1} & 2 \\ \end{array} } \right)\;\,\mathbf{S} = \mathbf{A}\;\mathbf{S} $$ and $\mathbf T$ is also a matrix of independent vectors, then it shall be possible to invert this relation $$ \mathbf{S} = \mathbf{A}^{\, - \,\mathbf{1}} \;\mathbf{T} $$ i.e. the matrix of the coefficients shall be invertible, i.e. its determinant shall be non null.
Viceversa if the determinant of $\mathbf{A} $ is not null, it is not possible to get a null vector by multiplying the matrix by a non-null vector.