Proof of existence of a vector with a property.

Question

So I tried to put the basis of $V$ in a matrix $A$ and multiply $A$ by $w$, but I can't find a way to make it satisfy the property.

Should I make $w = \frac{1}{v_n}$ and all the rest of the $v$ vectors elements of $ker(A)$?

Help, please?

Answer 1

The system$$\left\{\begin{array}{l}\langle w,v_1\rangle=0\\\langle w,v_2\rangle=0\\\vdots\\\langle w,v_{n-1}\rangle=0\end{array}\right.$$is a homogeneous sytem of $n-1$ linear equations. Therefore, it has some non-null solution $w$. We can't have $\langle w,v_n\rangle=0$ because then $w$ would be orthogonal to every element of $V$ and therefore orthogonal to itself. Multiplying $w$ by an appropriate constant, you get the vector that you're after.

Answer 2

For $n=2$ a solution would be $\vec w =\vec v_2-\frac{\langle\vec v_2,\vec v_1\rangle}{\langle\vec v_1,\vec v_1\rangle} \vec v_1$.

For $n>2$ the longer version of this approach would only work if $\langle\vec v_i,\vec v_j\rangle = 0$ for all $i,j$ with $1\le i<j<n$.