308F/G Winter 2013, Quiz 4

  1. The first problem was 27 from section 1.5. The answer is $ \left(\begin{array}{cc} 4 & 12 \\ 4 & 10 \end{array}\right)$.
  2. This was problem 39 from section 3.7. Let $x,y\in U$. Then because $F$ and $G$ are both linear, \[G\circ F(x+y)=G(F(x+y))=G(F(x)+F(y))=G(F(x))+G(F(y))=G\circ F(x)+G\circ F(y),\] and if $c\in \mathbb{R}$, \[G\circ F(cx)=G(F(cx))=G(cF(x))=cG(F(x))=cG\circ F(x)\] thus, $G$ is a linear transformation.