2
$\begingroup$

Consider vectors $w_1, w_2, w_3, w_4 \in \mathbb{R}^{n}$, $n \in \mathbb{Z}_{\geq 1}$.

Assume the following statement. For all $(c_1,c_2) \in (\mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0}) \setminus\{(0,0)\}$ there exists $x \in \mathbb{R}^n$ such that

$$ \left( c_1 w_1 + c_2 w_2 \right)^\top x < 0 \quad \text{ and } \quad \left( c_1 w_3 + c_2 w_4 \right)^\top x < 0. $$

Find a case in which the following statement is false. There exists $y \in \mathbb{R}^n$ such that

$$ w_i^\top y < 0 \quad \forall i \in \{1,2,3,4\}. $$

  • 0
    Maybe you could simplify your hypotheses just assuming that you're given vectors $w_1, w_2, w_3, w_4$? I mean, if there is no restriction on matrices $A_1, A_2$, I cannot find any restriction on vectors $w_i$ either.2012-08-31
  • 0
    @Adam: it would help if you showed what work you had already done, or why you couldn't do any.2012-08-31
  • 0
    Ok, I'll just use $w_i$s. I already found that if $A_1 = A_2$, then the first statement implies the second. Therefore I'm claiming that this implication becomes false whenever $A_1 \neq A_2$.2012-08-31

1 Answers 1