<example>

<example>
   <title>Linear Maps</title>
   <statement>
     <p>
       Let <m>P_n(\mathbb R)</m> denote the real vector space of polynomials
       of degree at most <m>n</m>.
       The map <m>T:P_n(\mathbb R) \to P_n(\mathbb R)</m> given by
       <m>T(p) = p'</m>, taking a polynomial to its derivative,
       is a linear map.
     </p>
   </statement>
   <hint>
     <p>
       What do you know about how the derivative affects the degree
       of a polynomial and what did you learn in calculus about the
       derivative of a sum of differentiable functions?
     </p>
   </hint>
   <hint>
     <p>
       How about the derivative of a constant times a differentiable function?
     </p>
   </hint>
</example>

<example>
  <title>Testing vector space axioms</title>
  <statement>
    <p>
      Polynomials of degree 3 do not form a vector space.
    </p>
  </statement>
  <answer>
    <p>
      The set is not closed under addition and lacks an additive identity.
    </p>
  </answer>
  <solution>
    <p>
      The sum of elements <m>x^3 + x^2</m> and <m>-x^3 + x</m> equals
      <m>x^2 + x</m> which is not of degree 3,
      so the set is not closed under vector addition.
    </p>
  </solution>
  <solution>
    <p>
      We can also note that the zero polynomial does not have degree 3,
      so is not an element of the set.
    </p>
  </solution>
</example>