0
$\begingroup$

Let $ x,y,z \in{\mathbb R}^3$. Show that $x,y \text{ and } z$ are linear independent iff $x \times y, x \times z \text{ and } y \times z$ are linear independent. Where $\times$ denotes the cross product in ${\mathbb R}^3.$

So far I have only been able to show the forward implication: If $x,y,z$ are linear dependent, then $x \times y, x \times z \text{ and } y \times z$ are also linear dependent.

I need help with the backwards implication: If $x \times y, x \times z \text{ and } y \times z$ are linear independent, then $x,y,z$ are linear independent

I would be grateful for some help.

$\endgroup$
1
$\begingroup$

Assume the cross products are linearly independent. Suppose $ax+by+cz=0$ so $ax\times y + cz\times y =0$. Hence $a=c=0$. Similarly you can prove $b=0$. Since no non-zero choices of $a,\,b,\,c$ work, $x,\,y,\,z$ are linearly independent.

$\endgroup$
0
$\begingroup$

\begin{eqnarray*} det \left( \begin{array}{ccc} a_2 b_3-a_3 b_2 & a_3 b_1-a_1 b_3 & a_1 b_2-a_2 b_1 \\ b_2 c_3-b_3 c_2 & b_3 c_1-b_1 c_3 & b_1 c_2-b_2 c_1 \\ c_2 a_3-c_3 a_2 & c_3 a_1-c_1 a_3 & c_1 a_2-c_2 a_1 \\ \end{array} \right)=\left( det \left( \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{array} \right) \right)^2 \end{eqnarray*}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.