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 $\dim S.$

• $x = 0.$
• 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 $A = (0, a_2, a_3)$ and $B = (0, b_2, b_3)$ be two elements in $S.$ Then, to satisfy closure under addition, $A + B$ must also be in $S.$ But since $A$ and $B$ are elements of $V_3,$ addition of elements occurs component-wise, giving us $$A + B = (0, a_2 + b_2, a_3 + b_3),$$ which is another element of $S.$ Thus, $S$ satisfies closure under addition. And if $c$ is a real scalar, then multiplication also occurs component-wise, giving us $$cA = c(0, a_2, a_3) = (0, ca_2, ca_3),$$ which is yet another element in $S,$ satisfying closure under multiplication by real numbers. Following Theorem 1.4, $S$ is a subspace of $V_3$ with dimension $2, spanned by $\{(0, 1, 0), (0, 0, 1)\}.$\quad \blacksquare$