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Didn't see this in any of the errata:

Problem 1-8(b): Suppose $T$ is a linear transformation. If there is a basis $x_1, \dotsc, x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \dotsc, \lambda_n$ such that $Tx_i = \lambda_i x_i$, prove that $T$ is angle preserving if and only if all $|\lambda_i|$ are equal.

Counterexample: the linear transformation taking $(0,1)$ to itself and $(1,1)$ to $-(1,1)$ does not preserve the angles between those two vectors, although it is diagonalizable with all eigenvalues of norm $1$.

Did Spivak perhaps mean to require that the $x_i$ be an orthogonal basis?

Eric Auld
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    There is indeed a mistake in this exercise. If I recall correctly, he originally meant $\lambda_i$ instead of $|\lambda_i|$. This is a related question. – Ian Mateus Jan 09 '15 at 23:02
  • @IanMateus Taking $\lambda_i$ instead of $\lvert \lambda_i \rvert$ is also problematic because the map $T(x,y) = (x,-y)$ has distinct eigenvalues ($1$ and $-1$) but is angle preserving, so the "only if" part then becomes false. –  Mar 24 '20 at 04:13

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