\\\\(
\nonumber
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\\\\)
Opg 1: Øvelsesopgavesæt
- Det oplyses, at $\, x(t) = t^3\,$ er løsning til følgende differentialligning, hvor $\,q(t)\,$ betegner en kontinuert funktion.
$$
x''(t) - x'(t) - 2x(t)=q(t) - t^2,\quad t\in\reel\,.$$
Find $\,q(t)\,$ og bestem samtlige løsninger til differentialligningen.
- Find rødderne i polynomiet
$$
z^2 + 2z + 5\,.$$
Angiv rødderne både på formen $a + ib$ og på formen $r_v$.
- En lineær afbildning $\,f:\reel^2\rightarrow\reel^\,3$ har følgende matrix med hensyn til en given basis $\,(\ma_1,\ma_2)\,$ i $\,\reel^2\,$ og en given basis $\,(\mv_1,\mv_2,\mv_3)\,$ i $\reel^3$.
$$
\mF=\begin{matr}{rrr} 1 & 4 \\\\ 2 & 5 \\\\ 3 & 6 \end{matr}\,.$$
Find matricen for $f$ med hensyn til basen $(\ma_1+\ma_2,\ma_1-\ma_2)$ i $\reel^2$ og den samme basis som før,
$(\mc_1, \mc_2, \mc_3)$ i $\reel^3$.