In Serge Lang's Linear Algebra textbook 3rd edition, there is an exercise in chapter 3.4, exercise 8 (a) which asks us to demonstrate that the linear map defined below is invertible.
Let $L: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the linear map as defined by $L(x, y, z) = (x - y, x + z, x + y +2z)$.
I'm wondering if there is an error. For example, $L(1, 1, -1)=(0, 0, 0)$, so the kernel is nonzero. Hence $L$ is not injective and therefore is not invertible.