Let V be a vector space over F and T in L(V,W). Prove that the following statements are equivalent (ie. a) implies b) and vice versa):

a) range(T) <intersection> null(T) = 0

b) If T(Tv)=0, then Tv=0, for v in V.

Thanks

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- Mar 8th 2013, 03:59 AMzachoonLinear algebra proof regarding linear maps
Let V be a vector space over F and T in L(V,W). Prove that the following statements are equivalent (ie. a) implies b) and vice versa):

a) range(T) <intersection> null(T) = 0

b) If T(Tv)=0, then Tv=0, for v in V.

Thanks - Mar 8th 2013, 04:55 AMHallsofIvyRe: Linear algebra proof regarding linear maps
Was this the complete statement of the problem? It doesn't quite make sense. You specify that V is a "vector space over F" but say nothing about W. Further, the range of T is a subspace of W while null(T) is a subspace of V. Are you assuming that V and W are the same? If not what do you mean by the "intersection" of two subspace of

**different**vector spaces?

Assuming that V= W, suppose v is in "null(T)". What does that tell you about v? Suppose w is in range(T). What does that tell you about w? - Mar 8th 2013, 06:42 AMzachoonRe: Linear algebra proof regarding linear maps
Whoops, yes this was mis-copied. let me write it out again.

Let V be a vector space over F and T in L(V,V). Prove that the following statements are equivalent (ie. a) implies b) and vice versa):

a) range(T) <intersection> null(T) = 0

b) If T(Tv)=0, then Tv=0, for v in V.

For this problem, i am still completely lost as to the correlation between the 2 statements. I can't percieve anything significant from the intersection of range(T) and null(T) to equal the set {0}, as it seems very general. - Mar 8th 2013, 08:33 AMHallsofIvyRe: Linear algebra proof regarding linear maps
Do you understand how "range(T)" and "null(T)" are defined? Suppose range(T) intersect null(T)= {0}. If T(T(v))= 0, that means that T(v) is in the null space, doesn't it? But obviously T(v) is in the range(T). Therefore T(v) is in both null(T) and range(T), therefore T(v)= what?

Now suppose that "if T(T(v))= 0, then Tv= 0" and further that u is in both range(T) and null(T). Since u is in null(T), what is T(u)? Since u is in range(T), there exist v such that T(v)= what? - Mar 8th 2013, 12:53 PMzachoonRe: Linear algebra proof regarding linear maps
thank you, that helped a lot!