Thread: Linear 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

Originally Posted by zachoon

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

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?

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.

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?

thank you, that helped a lot!