# Thread: Can anybody help me?

1. ## Can anybody help me?

I got two questions about proving linear mapping stuff. For questions number 5 and 6 in the attachment! ThanksMATH_136-Assign5-F09.pdf

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2. Remember that a function $\displaystyle f: \mathbb{R} ^n \rightarrow \mathbb{R}^m$ is linear iff for all $\displaystyle x,y \in \mathbb{R} ^n$ and $\displaystyle a,b \in \mathbb{R}$ we have $\displaystyle f(ax+by)=af(x)+bf(y)$. If you have $\displaystyle g,h: \mathbb{R} ^n \rightarrow \mathbb{R}^m$ the sum $\displaystyle (g+h)(x):=g(x)+h(x)$ and $\displaystyle (ag)(x):=ag(x)$. It should be clear from here, just try to work them out.

For the second one suppose $\displaystyle L: \mathbb{R} ^n \rightarrow \mathbb{R}^m$ and $\displaystyle M: \mathbb{R} ^m \rightarrow \mathbb{R}^p$ are linear take $\displaystyle x,y \in \mathbb{R} ^n$ and $\displaystyle a,b \in \mathbb{R}$ then $\displaystyle M \circ L(ax+by)=M(L(ax+by)=M(aL(x)+bL(y))=aM(L(x))+bM(L(y ))$