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 $\leq k,$ 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.
• Let $f$ and $g$ be elements of $S.$ More specifically, let $f$ and $g$ be given by \begin{align*} f(x) &= \sum_{j=0}^k a_jx^j, \quad g(x) = \sum_{j=0}^k b_jx^j \end{align*} Then, their sum $f + g$ 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 \end{align*} Which we can see is another polynomial of degree $\leq k,$ satisfying closure under addition. If we let $c$ be a real scalar, then \begin{align*} cf(x) &= c\sum_{j=0}^k a_jx^j \\ &= \sum_{j=0}^k ca_jx^j \end{align*} which is another polynomial of degree $\leq k,$ satisfying closure under addition. Thus, the set of polynomials of degree $\leq k$ including $f = 0$ is a subspace of $P_n.$
  
  To compute the dimension of $S,$ we note that $S$ is spanned by the independent set of $k + 1$ elements $\{1, x, \dots, x^k\},$ hence $\dim S = k + 1.\quad \blacksquare$