If there is a basis x1,...,xn of Rn and numbers λ1,...,λn such that Txi = λixi, prove that T is angle preserving if and only if all |λi| are equal.

To prove that the linear transformation \( T \) is angle-preserving if and only if all \( |\lambda_i| \) are equal, let's break down the problem step by step.

### Given:
- \( x_1, x_2, \ldots, x_n \) is a basis of \( \mathbb{R}^n \).
- \( T \) is a linear transformation such that \( T(x_i) = \lambda_i x_i \) for each \( i \).
- \( \lambda_1, \lambda_2,

Explore how to prove that a linear transformation preserves angles if and only if all eigenvalues are equal. This problem involves understanding basis vectors, eigenvalues, and their implications in linear algebra. Suitable for undergraduate students studying linear algebra.

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