How can I prove that a vector minus its average belongs to the orthogonal space C(1)?

Question

I stumbled onto a linear-algebra exercise that requires me to prove that:

Y - ȳ$\vec{1}$ belongs to the orthogonal space C(1) (vector space formed by the multiples of the vector 1).

I know that I have to be able to express Y - ȳ1 as a linear combination of other vectors whose all elements are the same (e.g: α₁ * [β₁ β₁ ... β₁] + α₂ * [β₂ β₂ ... β₂] + ... + α_n*[β_n β_n ... β_n]), but I'm clueless on how to do that.

Obs.1: ȳ is the mean of the elements in vector Y, and ȳ$\vec{1}$ is the multiplication of ȳ and the all-ones vector $\vec{1}$

Obs.2: I'm not sure if the proof that I want even makes sense, so if you can please explain why if it doesn't, that'd be great.

Thanks.

Answer 1

The precise statement is that $y-\overline{y}(1,1,...,1):=\left(y_1-\frac{y_1+...+y_n}{n},y_2-\frac{y_1+...+y_n}{n},...,y_n-\frac{y_1+...+y_n}{n}\right)$ is orthogonal to $(1,1,...,1)$.

You can see this by multiplying $\left(y-\overline{y}(1,1,...,1)\right)\cdot (1,1,...,1)=y\cdot(1,1,...,1)-\overline{y}(1,1,...,1)=(y_1+...+y_n)-n\overline{y}=0$