# Need clarification on Exercise 2.2.7 Hatcher

I am looking at Exercise 2.2.7 of Hatcher (pg 155):

For an invertible linear transformation $f: \Bbb{R}^n \to \Bbb{R}^n$ show that the induced map on $H_n(\Bbb{R}^n,\Bbb{R}^n - \{0\}) \approx \tilde{H}(\Bbb{R}^ n -\{0\}) \approx \Bbb{Z}$ is $\Bbb{1}$ or $\Bbb{-1}$ according to whether the determinant of $f$ is positive or negative.

Now what I don't understand in the question is the phrase "induced map on $H_n(\Bbb{R}^n,\Bbb{R}^n - \{0\})$. What does this mean? My interpretation is that $f: \Bbb{R}^n \to \Bbb{R}^n$ somehow gives a map $g$ from $H_n(\Bbb{R}^n,\Bbb{R}^n - \{0\})$ to itself.

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Here is a restatement: $f$ induces a map of the pair $(R^n, R^n-{0})$ to itself. The $n$th homology of this pair is isomorphic to Z. The induced map on this homology group is $\pm 1$ depending on the sign of the determinant of $f$.
I know that my map $f$ will induce a map between $H_n(\Bbb{R}^n$ and itself. However now for the pair of spaces, $(\Bbb{R}^n,\Bbb{R}^n - \{0\})$ isn't the induced map ambiguous? – user38268 Oct 6 '12 at 4:15
In particular, $f$ induces a map from $\mathbb{R}^n \setminus {0}$ to itself because $f$ is invertible. – Adam Saltz Oct 6 '12 at 13:32