Find a basis $ {\{b_1,\cdots, b_n}\} $ of $ \mathbb{C}^n $ such that $ \langle b_j, b_k \rangle = 1 $ whenever $ j \neq k $

Question

Find a basis $ {\{b_1,\cdots, b_n}\} $ of $ \mathbb{C}^n $ such that $ \langle b_j, b_k \rangle = 1 $ whenever $ j \neq k $ where $\left\langle (x_1,\cdots,x_n),(y_1,\cdots,y_n)\right\rangle:=\sum_{k=1}^{n}\overline{x_k}y_k.$

Friends, I have not been able to solve this problem, I feel like I should use the Gram-Schmidt orthonormalization process, correct? Could you give me a hint?

Answer 1

Hint: start with an orthonormal basis, and then add one basis vector to all the others.