In Exercises 1 through 28, determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The functions in Exercises 1 through 17 are real-valued. In Exercises 3, 4, and 5, each function has domain containing 0 and 1. In Exercises 7 through 12, each domain contains all real numbers.
23. All vectors $(x, y, z)$ in $V_3$ with $x = 0$ or $y = 0$.
Let $X = (0, x_2, x_3)$ and $Y = (y_1, 0, y_3)$ where $x_2$ and $y_1$ are nonzero. We can see that $X$ and $Y$ are members of the set, but the sum $X + Y = (y_1, x_2, x_3 + y_3)$ is not. Since it fails to satisfy closure under addition, the set of all vectors $(x, y, z)$ in $V_3$ with $x = 0$ or $y = 0$ is not a real linear space. $\,\blacksquare$
