Mathematical Immaturity

1.10 Exercises

Let $P_n$ denote the linear space of all real polynomials of degree $\leq n,$ where $n$ is fixed. In each of Exercises 11 through 20, let $S$ denote the set of all polynomials $f$ in $P_n$ satisfying the condition given. Determine whether or not $S$ is a subspace of $P_n.$ If $S$ is a subspace, compute $\dim S.$

• $f$ has degree $k,$ where $k \lt n$ or $f = 0.$
• 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.
• $\text{Counterexample.}\quad$ Let $f$ and $g$ be defined as follows \begin{align*} f(x) &= \sum_{j=0}^k a_jx^j, \quad g(x) = \sum_{j=0}^k b_jx^j \end{align*} Now, suppose $b_k = -a_k.$ Then, the sum $(f + g)(x)$ is \begin{align*} (f+g)(x) &= f(x) + g(x) \\ &= \sum_{j=0}^k a_jx^j + \sum_{j=0}^k b_jx^j \\ &= \sum_{j=0}^k (a_j + b_j)x^j \\ &= \sum_{j=0}^{k-1} (a_j + b_j)x^j + (a_k - a_k)x^k \\ &= \sum_{j=0}^{k-1} (a_j + b_j)x^j \end{align*} Which we can see is a polynomial of degree $\leq k - 1,$ and not an element of $S.$ Thus, $S$ violates closure under addition and is not a subspace of $P_n.\quad\blacksquare$