Question

v1; v2; v3; v4 are linearly independent vectors in a vector space V .
Determine if {v1 - v2; v2 + 3v3; v1 - v3 + v4; v2 + v4g is linearly independent.

Answer 1

You can write your basis as
\[[v_1 v_2 v_3 v_4]\left[\begin{matrix}1 & 0 & 1 & 0\\-1 & 1 & 0 & 1\\0 & 3 & -1 & 0\\0 & 0 & 1 & 1 \end{matrix}\right]\]
where the v's are independent column vectors.
The product of the 2 matrices must still result in 4 independent vectors if you have independence. So do elimination on the right hand matrix to determine if it has 4 pivots, and so 4 independent vectors