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.
25. All vectors $(x, y, z)$ in $V_3$ with $3x + 4y = 1$, $z = 0$.
Let $A = (\frac{1}{3}, 0, 0)$ and $B = (0, \frac{1}{4}, 0).$ As we can see, they are vectors $(x, y, z)$ in $V_3$ with $3x + 4y = 1$ and $z = 0.$ But, $A + B = \left(\frac{1}{3}, \frac{1}{4}, 0\right)$ is a vector $(x, y, z)$ in $V_3$ with $3x + 4y \neq 1,$ violating closure under addition. Moreover, if we let $c$ be some real scalar we find that for $cA$ and $cB,$ $3x + 4y = 1$ is only satisfied when $c = 1,$ violating closure under multiplication by real numbers. Thus, the set of all vectors $(x, y, z)$ in $V_3$ with $3x + 4y = 1$ and $z = 0$ is not a linear space. $\,\blacksquare$
