Question

If u = (1, 1, 0) ; v = (1, 4, 1) ; w = (r^2, 1, r^2) . For which values of r, is the set (u,v,w) linearly independent?

Answer 1

\(\vec u\), \(\vec v\), and \(\vec w\) are linearly independent if the determinant of the matrix\[A=\left(\begin{matrix}\vec u_i&\vec u_j&\vec u_k\\\vec v_i&\vec v_j&\vec v_k\\\vec w_i&\vec w_j&\vec w_k\end{matrix}\right)\]is not zero.
In other words when \(\det(A)\neq0\), the vectors are linearly independent.
Find the roots of \(\det(A)=0\); all values besides these will lead to linear independence.