Orthogonalizing a basis into the standard basis of $\Bbb{R^n}$

Question

The problem is to orthogonalize the following basis in $\Bbb{R^n}$. $$e'_1=(1,0,...,0)$$$$e'_2=(1,1,...,0$$ $$\cdots$$ $$e'_n=(1,1,...,1).$$

Lets denote the orthogonal basis elements with $e_i$. Let $e'_1 = e_1$. Straightforward application of Gram-Schmidt method will yield vectors $$e_1 = ( 1,0,...,0)$$ $$e_2 = (0,1,...,0)$$ $$e_3 = (0,0,1,...,0)$$

and so on. Since we dont know what $n$ is, we need to prove that $$e_n = e'_n - e'_{n-1}$$

or other arguments to that end. Unfortunately, the previous equation doesn't seem to yield itself well to induction. Any advice?

Answer 1

By inspection,

$$e_n'=\sum_{k=1}^n e_k$$

Therefore, we have

$$e'_{n}-e'_{n-1}=e_n$$

And we are done!