1. [SOLVED] vectors and matrices

need immediate help

a) Let T: V V be a linear transformation, where V is a finite dimensional vector space. If dim(range(T))=dim(range(T2)) show that the range and null space of T only have the zero vector in common.
b)
Let T: l2 (Zn)--- l2 (Zn) be a linear transformation. Show that T is translation invariant if and only if T(z) = ∑K=0N-1 ak Rk(z) for some a0 ,……,aN-1

2. Originally Posted by wizard005
need immediate help

a) Let T: V V be a linear transformation, where V is a finite dimensional vector space. If dim(range(T))=dim(range(T2)) show that the range and null space of T only have the zero vector in common.
b)
Let T: l2 (Zn)--- l2 (Zn) be a linear transformation. Show that T is translation invariant if and only if T(z) = ∑K=0N-1 ak Rk(z) for some a0 ,……,aN-1
a) Do this by contradiction. Suppose the result is false, so there is a nonzero vector z in the range of T (so z=Tx for some x in V), with Tz=0. From that, you should be able to show that the null space of T^2 is bigger than the null space of T. Then use the "rank + nullity" theorem to deduce that the range of T^2 is smaller than the range of T.

b) This is so badly formatted that I can't read the question. It looks as though it should be saying something like

Let $\displaystyle T: l_2 (Z^n)\to l_2 (Z^n)$ be a linear transformation. Show that T is translation invariant if and only if $\displaystyle T(z) = \sum_{k=0}^{n-1} a_k R_k(z)$ for some $\displaystyle a_0 ,\ldots,a_{n-1}$.

Is that right? If so, what is $\displaystyle R_k(z)$?