Abstract Linear Algebra Question See an attached image I don

Abstract Linear Algebra Question. See an attached image. I don\'t want the proof that those statements are equivalent for V is finite-dimensional, I already did that. I want an example to show that that those statements would be false if V was infinite-dimensional.

Suppose V is finite-dimensional and T E L(V) the following are equivalent: (a) V = null T range T. (b) V-null T + range T. (c) null Tn range T={0}. Give an example to show that the exercise above is false without the hypothesis that V is finite-dimensional.

Solution

because T is a set of linearly independent list of vector, they have a basis.

nullT and rangeT are finite dimensional which when directly added or added or intersected would result in aunique finite dimension.