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.
24. All vectors $(x, y, z)$ in $V_3$ with $y = 5x$.
Let $A = (x_1, 5x_1, z_1)$ and $B = (x_2, 5x_2, z_2)$ be vectors $(x, y, z)$ in $V_3$ with $5 = 5x.$ Then,
\begin{align*}
A + B &= (x_1 + x_2, 5x_1 + 5x_2, z_1 + z_2)
\\
&= [x_1 + x_2, 5(x_1 + x_2), z_1 + z_2]
\end{align*}which is yet another vector $(x, y, z)$ in $V_3$ satisfying $y = 5x,$ thus verifying closure under addition. We can also see that since $A$ and $B$ have real-valued components, $A + B = B + A,$ verifying the commutative property of addition. If $C = (x_3, 5x_3, z_3),$ we have
\begin{align*}
(A + B) + C &= [(x_1 + x_2) + x_3, 5(x_1 + x_2) + 5x_3, (z_1 + z_2) + z_3]
\\
&= [x_1 + (x_2 + x_3), 5x_1 + 5(x_2 + x_3), z_1 + (z_2 + z_3)]
\\
&= A + (B + C)
\end{align*}satisfying the associative law of addition.
Now, let $c$ and $d$ be real scalars. Then, $cA = (cx_1, 5cx_1, cz_1),$ which is also a vector $(x, y, z)$ in $V_3$ satisfying $y = 5x,$ verifying closure under addition. The zero vector $O = (0, 0, 0)$ trivially satisfies $y = 5x$ and also satisfies $A + O = A$ for all $A$ in $V_3,$ satisfying the existence of the zero element. Setting $c = 1$ we can see that the set satisfies the existence of identity, and setting $c = -1,$ we see that it satisfies the existence of negatives.
With $A$ and $B$ as defined above, we have
\begin{align*}
c(dA) &= c(dx_1, 5dx_1, cz_1)
\\
&= (cdx_1, 5cdx_1, cdz_1)
\\
&= (cd)A
\\
\\
c(A + B) &= c(x_1 + x_2, 5x_1 + 5x_2, z_1 + z_2)
\\
&= (cx_1 + cx_2, 5cx_1 + 5cx_2, cz_1 + cz_2)
\\
&= cA + cB
\\
\\
(c + d)A &= (c + d)(x_1, 5x_1, z_1)
\\
&= [(c + d)x_1, (c + d)5x_1, (c + d)z_1]
\\
&= (cx_1 + dx_1, 5cx_1 + 5dx_1, cz_1 + dz_1)
\\
&= cA + dA
\end{align*}
This verifies that the set of vectors $(x, y, z)$ in $V_3$ verifies the ten axioms and is thus a linear space. $\,\blacksquare$
