Question

Determine whether or not the following vectors span
u1 = ( 1, 1, 2), u2 = ( 1, -1, 2 ) and u3 = ( 1, 0, 1 ).

Answer 1

Taking into account that $\left|\begin{array}{ccc} 1 & 1 & 2 \\ 1 & -1 & 2 \\ 1 & 0 & 1\end{array}\right |= -1+2-(-2+1)=1+1=2\ne 0,$ we conclude that the vectors $u_1 = ( 1, 1, 2), u_2 = ( 1, -1, 2 )$ and $u_3 = ( 1, 0, 1 )$ are linearly independent, and consequently span $\mathbb R^3.$