1. linearly independent problem

Let $\displaystyle u_{1}, u_{2}$ be nonzero vectors in $\displaystyle R^{n}$ such that $\displaystyle u_{1}u^{T}_{2}=0$.

Show that $\displaystyle u_{1}, u_{2}$ are linearly independent.

I let $\displaystyle u_{1}=(a_{1} a_{2} ... a_{n})$ and $\displaystyle u_{2}=(b_{1} b_{2} ... b_{n})$.

Then $\displaystyle u_{1}u^{T}_{2}=a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n }=0$.

How to continue from here?

2. Originally Posted by deniselim17
Let $\displaystyle u_{1}, u_{2}$ be nonzero vectors in $\displaystyle R^{n}$ such that $\displaystyle u_{1}u^{T}_{2}=0$.

Show that $\displaystyle u_{1}, u_{2}$ are linearly independent.

I let $\displaystyle u_{1}=(a_{1} a_{2} ... a_{n})$ and $\displaystyle u_{2}=(b_{1} b_{2} ... b_{n})$.

Then $\displaystyle u_{1}u^{T}_{2}=a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n }=0$.

How to continue from here?

Suppose $\displaystyle u_1,u_2$ are lin. depen. $\displaystyle \Longleftrightarrow u_1=\lambda u_2\,,\,\,0\neq\lambda\in\mathbb{R}$ , but then $\displaystyle 0=u_1u^T_2=(\lambda u_2)u^T_2=\lambda\left(u_2u^T_2\right)\Longrightar row u_2=0$ , contradiction.

Tonio