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Let P2 be a vector space of real polynomial functions of degree at most 2. Which of the following operations for polynomial functions define the transformation P2⟼P2? Wich are linear transformations?

  1. f: p ⟼f(p) with f(p): t ⟼t ∙ p(t)
  2. g: p ⟼g(p) with g(p): t ⟼t ∙ p'(t)
  3. h: p ⟼h(p) with h(p): t ⟼p(t)^2
  4. k: p ⟼k(p) with k(p): t ⟼p(t)+1

Okay, so I don't get it at all... I just can't figure out what "t ⟼t∙p(t)" etc. is supposed to mean. Would be great if somebody could help!

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  • $\begingroup$ Please use MathJax to format your question. $\endgroup$
    – Marconius
    Nov 5 '15 at 13:07
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$f(p)= t\cdot p(t)$ means t times the polynomial. For example if $p(t)= 3t^2- 3t+ 5$ then $f(p(t))= t\cdot p(t)= 3t^3- 3t^2+ 5t$. It should be obvious why this is NOT from P2 to P2.

$g(p)= t\cdot p'(t)$ means t times the derivative of the polynomial. If $p(t)= 4t^2- 3t+ 2$, then $p'(t)= 8t- 3$ so $g(p)= t(8t- 3)= 8t^2- 3t$. That does map quadratic polynomials to quadratic polynomials. Such a map is linear if and only if g(ap+ bq)= ag(p)+ bg(q) for any quadratic polynomials p and q. For example, with p as above and $q(t)= t^2- 2t+ 4$, $g(t)= t(2t- 2)= 2t^2- 2t$ so that $g(p)+ g(q)= (8t^2- 3t)+ (2t^2- 2t))= 10t^2- 5t$ while $p(t)+ q(t)= (4t^2- 3t+ 2)+ (t^2- 2t+ 4)= 5t^2- 5t+ 6$ so that $g(p+ q)= t(10t- 5)= 10t^2- 5t$. Those are the same. However, while a single counterexample would be sufficient to show a general statement is false, you would have to give a general proof that "$g(ap+ bq)= t(ap+ bq)'= ag(p)+ bg(q)= at(p'(t))+ bt(q'(t))$.

It helpful here to know that "differentiation" is itself a linear operation! That is, that (af+ bg)'= af'+ bg' for f and g any differentiable functions.

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