correcting a mistake in Spivak Spivak's Calculus on Manifolds asks the reader to prove this (problem 1-8, pp.4-5):


  If there is a basis $x_1, x_2, ..., x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \lambda_2, ..., \lambda_n$ such that $T(x_i) = \lambda_i x_i$, $1 \leq i \leq n$, prove that $T$ is angle-preserving iff $\left| \lambda_i \right| = c, 1 \leq i \leq n$.


Here "angle-preserving" means that the linear map $T$ satisfies $$\frac{ \langle x, y \rangle}{\|x\| \|y\|} = \frac{ \langle T(x), T(y) \rangle}{\|T(x)\| \|T(y)\|},$$
and that $T$ is injective.

My first problem with this question is that the claim is false. Taking $n = 2$, $x_1 = (1, 0)$, $x_2 = (1,1)$, $T(x_1) = -x_1$, $T(x_2) = x_2$, and setting $x = x_1$, $y = x_1 + x_2$, the expression in the RHS above evaluates to $0$, while the expression in the LHS evaluates to $\frac{1}{\sqrt{2}}$.

My second, bigger problem is that I'm not really understanding what's going on. An earlier part of the problem had me show that norm-preserving matrices are angle-preserving; this I'm not sure I get. Thus, I'm not sure what true "version" of this statement the author had in mind (was he trying to get a converse?) and I don't know what to do.



Here's my guess:

Looking at some transformations in $\mathbb{R}^2$ (just drawing them), it looks like some of them could be "flip the sign of a basis vector" IF it's an orthogonal basis. However, I can't seem to recover the usual "rotation through an angle $\theta$" transformation this way, so I'm not sure that requiring the basis be orthogonal makes the statement true. 

Also, I'm not sure how to take the inner product of vectors which aren't in standard coordinates. Or am I missing something here?
You are correct. This is one of the few errors in Spivak's Calculus on Manifolds. For this particular exercise, see the following questions:


Question about Angle-Preserving Operators
Action of angle-preserving linear transformation on basis vectors


copper.hat's answer to the first question cited in the above used the same counterexample as yours.

For your second question, I don't think there is any conflict. Spivak simply meant that if $T$ is a norm-preserving map, then it automatically enjoys the property of being angle-preserving. He did not say that the converse is true. In other words, norm preservation is a stronger condition than angle preservation. In fact, $T$ is norm-preserving map if and only if its matrix w.r.t. the standard basis is a real orthogonal matrix $Q$, and $T$ is angle-preserving if and only if its matrix w.r.t. the standard basis is $\lambda Q$ for some scalar $\lambda&gt;0$ and some real orthogonal matrix $Q$ (for a proof, see the aforementioned answer by copper.hat).
I noticed this too. If you ignore the absolute value on lambda, it's a true statement which can be easily proven.