# linear transformation sequences

I'm working on this exercize about linear transformation:

Let $E=\mathbb{R^N}$, $T:E \rightarrow E: \ (u_n)_{n\geq 0} \rightarrow (u_{n+1})_{n\geq 0}$

$S:E \rightarrow E: \ (u_n)_{n\geq 0} \rightarrow (v_{n})_{n\geq 0}$ with $v_0=0$ and for all $i \geq 1, v_i=u_{i-1}$

1)Prove T and S are linear transformation and tell if they are injective or surjective.

I prove T is a linear transformation using $T(\lambda (u_n)+(v_n))=\lambda T((u_n))+T(v_n))$ but how could we do for S according the the conditions ? How could we determine if they are injective or surjective ?

Thank you

## 1 Answer

$S((u_n)+(u'_n)) =S((u_n+u'_n)) = (w_n)_{n\geq0}$ where $w_n=0$ for and for $i>0$ $w_i=u_i+u'_i$. But then $(w_n)_{n\geq0}=(v_n)_{n\geq0}+(v'_n)_{n\geq0}$ where $v_0=0$ and $v_i=u_i$ for $i>0$ and $v'_0=0$ and $v'_i=u'_i$ for $i>0$. Thus $(w_n)_{n\geq0}=S(u_n)+S(u'_n)$. Thus $S((u_n)+(u'_n))=S(u_n)+S(u'_n)$. Show $S(\lambda(u_n)_{n\geq0})=\lambda S((u_n)_{n\geq0})$ the same way.