It is my homework from Hatcher's book.
It is a problem 7 on section 2.2, stating:
For an invertible linear transformation $f:\mathbb{R}^n \to \mathbb{R}^n$ show that the induced map on $H_n (\mathbb{R}^n, \mathbb{R}^n-{0}) \sim H_{n-1} (\mathbb{R}^n-{0}) \sim \mathbb{Z} $ is identity or -identity according to whether the determinant of $f$ is positive or negative.
Since $f$ is homeomorphism, it seems obvious that induced homomorphism should be identity or -identity. However, I have no idea how it can be connected to the deterimant of a map.
Any comment would be grateful! Thank you for reading my question!