Question

Show whether the first three vectors are linearly independent

V1= 1, 1, 2, -2

V2= 2, -3, 0, 2

V3= -2, 0, 2, 2

V4= 3, -3 -2, 2

Answer 1

Consider the matrix $A=\left(\begin{array}{cccc}1 & 1 & 2 & -2\\ 2 & -3 & 0 & 2\\-2 & 0 & 2 & 2\end{array}\right)$ that contains the rows the vectors $v_1, v_2,v_3.$ Since $A=\left|\begin{array}{ccc}1 & 1 & 2 \\ 2 & -3 & 0 \\-2 & 0 & 2 \end{array}\right|=-6-12-4=-22\ne 0$, we conclude that $rank(A)=3$, and therefore, the vectors $v_1, v_2,v_3$ are linearly independent.