Question

1. For what values of h the vectors

⟶ [ 1 ⟶ [ -1 ⟶ [ 5

u₁ = -3 u₂ = 9 u₃ = -7

-2 ] -6 ] h ]

are linearly independent? (Show all working)

Answer 1

$\vec{a}=(1,-1,5), \vec{b}=(-3,9,-7), \vec{c}=(-2,-6,h)\\ \lambda_1\vec{a}+\lambda_2\vec{b}+\lambda_3\vec{c}=\vec{0}\\ \lambda_1=\lambda_2=\lambda_3=0\\ \lambda_1(1,-1,5)+\lambda_2(-3,9,-7)+\lambda_3(-2,-6,h)=\\=(0,0,0)\\ \lambda_1-3\lambda_2-2\lambda_3=0\\ -\lambda_1+9\lambda_2-6\lambda_3=0\\ 5\lambda_1-7\lambda_2+h\lambda_3=0\\ \Delta= \begin{vmatrix} 1& -3&-2 \\ -1 & 9&-6\\ 5&-7&h \end{vmatrix}=9h+90-14+\\+90-3h-42=6h+124\neq0\\ h\neq-\frac{124}{6}\\ h\neq-\frac{62}{3}\\$

If $h\neq-\frac{62}{3}, h\in(-\infty,-\frac{62}{3})\cup(-\frac{62}{3},\infty)$ the vectors $\vec{a},\vec{b},\vec{c}$ are linearly independent.