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.
22. All vectors $(x, y, z)$ in $V_3$ with $z = 0$.
Let $A = (x_0, y_0, 0)$ and $B = (x_1, y_1, 0)$ be vectors in $V_3$ with $z = 0.$ Then, $A + B = (x_0 + x_1, y_0 + y_1, 0)$ is a vector $(x, y, z)$ in $V_3$ with $z = 0.$ If $c$ is a real scalar, then $cA = (cx_0, cy_0, 0)$ is also a vector $(x, y, z)$ in $V_3$ with $z = 0.$ As such, we have verified the closure axioms.
The vector $O = (0, 0, 0)$ satisfies the existence of the zero element since it is a vector $(x, y, z)$ in $V_3$ with $z = 0$ such that $A + O = A$ for all $A$ in the set. If we let $c = 1,$ we can see that the set satisfies the existence of identity and if we set $c = -1$ we find that $A + cA = A - A = O,$ verifying the existence of negatives.
We can confirm element-wise that for any real vectors $A,$ $B,$ and $C$ in the set, we have $A + B = B + A$ and $(A + B) + C = A + (B + C),$ thus satisfying the commutative and associative laws of addition.
To verify the axioms for multiplication by real numbers, let $a$ and $b$ be real scalars and let $X = (x_1, x_2, 0)$ and $Y = (y_1, y_2, 0)$ Then, $bX = (bx_1, bx_2, 0)$ and $a(bX) = a(bx_1, bx_2, 0) = (abx_1, abx_2, 0) = (ab)X.$ Since addition of vectors in $V_3$ occurs element-wise, we have $(X + Y) = (x_1 + y_1, x_2 + y_2, 0)$ with
\begin{align*}
a(X + Y) &= a(x_1 + y_1, x_2 + y_2, 0)
\\
&= (ax_1 + ay_1, ax_2 + ay_2, 0)
\\
&= aX + aY
\end{align*}
And since the components of any $X$ in $V_3$ are presumed to be real-valued, we can see that
\begin{align*}
(a + b)X &=\left[(a + b)x_1, (a + b)x_2, 0\right]
\\
&=\left(ax_1 + bx_1, ax_2 + bx_2, 0\right)
\\
&= aX + bX
\end{align*}
Thus, we have shown that the set of all vectors $(x, y, z)$ in $V_3$ with $z = 0$ is a real linear space. $\,\blacksquare$
