Vector orthogonal to linear independent set of vectors is not in their span

Question

Suppose I have a set of linear independent vectors $v_1.....v_n$, and suppose now I have a vector $w$ that is orthogonal to all of them.

I would like to prove that $w$ is not in the span of these vectors.

I am a little embarrassed to say that I don't quite know a "clean" way to prove this, in that, without first proving that the following:

if $w$ is in the span of $v_1....v_n$, that is $w = a_1 v_1 ......a_n v_n$ and assume WLOG $a_1 \neq 0$, then we can express $w$ in terms of an orthogonal basis of its span (By Gram-Schmidt), $v_1,u_2.....u_n$ where $w = a_1 v_1 + .....b_n u_n$ and then showing that the dot product of $w$ with $v_1$ in this case equals to $a_1$ which is a contradiction.

I am unable to prove this problem with just the linearly independent assumption, can someone offer a clean proof of it?

Answer 1

Hint:

Assume that $w$ İS in their span, i.e $w$ can be written as a linear combination of vectors $v_1, ..., v_n$.

Can you show that $w$ cannot be orthogonal to the vectors $v_1,...,v_n$ at the same time ?

Answer 2

Say $w\in\text {span}\{v_1,\dots,v_n\}$. Then $w\cdot w=0\implies w=0$.