Mathematical Immaturity

1.10 Exercises

In each of Exercises 1 through 10, let $S$ denote the set of all vectors $(x, y, z)$ in $V_3$ whose components satisfy the condition given. Determine whether $S$ is a subspace of $V_3.$ If $S$ is a subspace, compute $\dim S.$

• $y = 2x$ and $z = 3x.$
• Recall from Section 1.6, Theorem 1.4, that a nonempty subset $S$ of a linear space $V$ is a subspace if and only if it satisfies the closure axioms:
  
  $\text{Axiom 1.}\quad$Closure under addition. $\quad$ For every pair of elements $x$ and $y$ in $V$ there corresponds a unique element in $V$ called the sum of $x$ and $y,$ denoted by $x + y.$
  $\text{Axiom 2.}\quad$Closure under multiplication by real numbers. $\quad$ For every $x$ in $V$ and every real number $a$ there corresponds an element in $V$ called the product of $a$ and $x,$ denoted by $ax.$
  
  The dimension of a basis for a linear space is the number of elements in the basis.
• Let $A = (a, 2a, 3a)$ and $B = (b, 2b, 3b)$ be two elements of $S,$ where $a$ and $b$ are real scalars. Then, their sum $A + B$ is \begin{align*} A + B &= (a + b, 2a + 2b, 3a + 3b) \\ &= \left[a + b, 2(a + b), 3(a + b)\right] \end{align*} from which we can see that $y = 2x$ and $z = 3x,$ satisfying closure under addition. If we let $c$ be some real scalar, we find that \begin{align*} cA &= c(a, 2a, 3a) \\ &= (ca, 2ca, 3ca) \end{align*} which satisfies closure under multiplication. Thus, $S$ is a subspace of $V_3.$ To find the dimension of $S,$ we note that $S$ is spanned by the set $\{(1, 2, 3)\}$ thus giving $\dim S = 1.\quad\blacksquare$