Can we generalize Pythagorean theorem to more than $2$ vectors?

Question

In Linear algebra done right: the statement of pythagorean theorem is:

Suppose u and v are orthogonal vectors in V, then $||u+v||^{2}=||u||^{2}+||v||^{2}$

And can we generalize the statement to:

Suppose $u_{1},...,u_{n}$ are orthogonal vectors in V, then $||u_{1}+...+u_{n}||^{2}=||u_{1}||^{2}+...+||u_{n}||^{2}$

If it is true, could someone please tell me how to prove it? If it is false, could someone tell me why?

Answer 1

We know it is true for two orthogonal vectors $v, w \in V$.

Then, let $v_1=u_1+\dots+u_{n-1}$, we have $$||u_1+\dots+u_n||^2=||v+u_n||^2=||v_1||^2+||u_n||^2=||u_1+\dots+u_{n-1}||^2+||u_n||^2$$ because $v_1$ and $u_n$ are orthogonals.

Repeating the argument with $v_2=u_1+\dots+u_{n-2}$ and $u_{n-1}$ to obtain $$||u_1+\dots+u_n||^2=||v_2+u_{n-1}||^2+||u_n||^2=||u_1+\dots+u_{n-2}||^2+||u_{n-1}||^2+||u_{n}||^2.$$

Iterating this argument we can easily prove the thesis by induction.

You can look here for more details.

Answer 2

It is true. You can easily prove it by induction.