If $¯ u$ $bar u$ , $¯ v$ $bar v$ , and $¯ ¯ ¯ w$ $bar w$ are linearly independent vectors, find the values of $t$ $t$ ?

Question

$(t^{2} - t - 2) ¯ u + (2 t^{2} - 3 t - 2) ¯ v + (3 t^{2} - 5 t - 2) ¯ ¯ ¯ w = 0$ $(t^2-t-2)baru+(2t^2-3t-2)bar v+(3t^2-5t-2)bar w=0$

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Answer 1 · 2018-05-11T08:41:00

$t = 2$ $t = 2$

If these vectors are linearly independent, then:

$α u + β v + γ w = 0 \Rightarrow α, β, γ = 0$ $alpha bb u + beta bb v + gamma bb w = bb 0 implies alpha, beta, gamma = 0$

Or in this case:

Factoring:

$α (t) = t^{2} - t - 2 = (t - 2) (t + 1)$ $alpha(t) = t^2-t-2 = (t-2)(t+1)$
$β (t) = 2 t^{2} - 3 t - 2 = (t - 2) (2 t + 1)$ $beta(t) = 2t^2-3t-2= (t - 2) (2 t + 1)$
$γ (t) = 3 t^{2} - 5 t - 2 = (t - 2) (3 t + 1)$ $gamma(t) = 3t^2-5t-2 = (t - 2) (3 t + 1)$

$\Rightarrow t = 2$ $implies t = 2$