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?
Question about Angle-Preserving Operators This an exercise out of Spivak's "Calculus on Manifolds".  Edit:  There was a typo in the exercise as is noted below in the answers.  The statement has been edited to reflect this.  Given $x,y\in\mathbb{R}^{n}$, the angle between $x$ and $y$ is defined by  $$\angle(x,y) = \arccos\left(\frac{\langle x,y \rangle}{|x|\cdot |y|}\right),$$ where $\langle x,y \rangle$ denotes the standard Euclidean inner product.  A linear operator $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is said to be angle-preserving if $\angle(T(x),T(y)) = \angle(x,y)$ for every $x,y\in\mathbb{R}^{n}$.  The exercise as stated:  Let $\{x_{1},\dots, x_{n}\}$ be a basis for $\mathbb{R}^{n}$.  Then suppose that $\lambda_{1}, \dots, \lambda_{n}\in \mathbb{R}$ are such that $Tx_{j} = \lambda_{j}x_{j}$ for each $j = 1,\dots, n$.  Then $T$ is angle-preserving only if (not if and only if!)$|\lambda_{i}| = |\lambda_{j}|$ for every $1\leq i\leq j\leq n$.  I'm having problems with the $(\Rightarrow)$ direction.  My best attempt (which seems to lead nowhere) is to suppose that $|\lambda_{j}|\neq |\lambda_{k}|$.  Then by assumption, \begin{align*} \angle(Tx_{j},Tx_{k}) &amp; = \arccos\left(\frac{\langle Tx_{j},Tx_{k} \rangle}{|Tx_{j}|\cdot |Tx_{k}|}\right)\\ &amp; = \arccos\left(\frac{\langle \lambda_{j}{x_{j}},\lambda_{k}{x_{k}} \rangle}{|\lambda_{j}{x_{j}}|\cdot |\lambda_{k}{x_{k}}|}\right)\\ &amp; = \arccos\left(\frac{\lambda_{j}\lambda_{k}\langle {x_{j}},{x_{k}} \rangle}{|\lambda_{j}|\cdot|\lambda_{k}|\cdot|{x_{j}}|\cdot |{x_{k}}|}\right)\\ &amp; = \arccos\left(\text{sign}(\lambda_{j})\text{sign}(\lambda_{k})\frac{\langle {x_{j}},{x_{k}} \rangle }{|{x_{j}}|\cdot |{x_{k}}|}\right)\\ \end{align*} may also be calculated as \begin{align*} \angle(Tx_{j},Tx_{k}) &amp; =  \angle(x_{j},x_{k})\\ &amp; = \arccos\left(\frac{\langle x_{j},x_{k} \rangle}{|x_{j}|\cdot |x_{k}|}\right). \end{align*}  Then since $\arccos$ is injective, I believe I can make the jump that $\text{sign}(\lambda_{j})\text{sign}(\lambda_{k}) = 1$, which does not resemble the conclusion that I should arrive at.  Note:  I wasn't sure what tag to put this under, so anyone who knows better please feel free to adjust.  Thanks for any help you can give.
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