Question

1. For what values of h the vectors

⟶ [ 1 ⟶ [ -1 ⟶ [ 5

u = -3 u_{2 =} 9 u₃ = -7 <--- u₁, u₂, and u₃ are vectors

-2] -6] h]

are linearly independent?

Answer 1

$\vec{x_1}=(1,-1,5), \vec{x_2}=(-3,9,-7), \vec{x_3}=(-2,-6,h)\\ k_1\vec{x_1}+k_2\vec{x_2}+k_3\vec{x_3}=\vec{0}\\ k_1=k_2=k_3=0\\ k_1(1,-1,5)+k_2(-3,9,-7)+k_3(-2,-6,h)=\\=(0,0,0)\\ k_1-3k_2-2k_3=0\\ -k_1+9k_2-6k_3=0\\ 5k_1-7k_2+hk_3=0\\ \Delta= \begin{vmatrix} 1& -3&-2 \\ -1 & 9&-6\\ 5&-7&h \end{vmatrix}=9h+90-14+\\+90-3h-42=6h+124\neq0\\ h\neq-\frac{124}{6}\\ h\neq-\frac{62}{3}\\$