- #1

- 1,357

- 0

**[SOLVED] Dimensionality, Rangespace & Nullspace Problem**

**Homework Statement**

Prove (where A is an n x n matrix and so defines a transformation of any n-dimensional space V with respect to B, B where B is a basis of V) that [itex]\dim(R(A) \cap N(A)) = \dim R(A) - \dim R(A^2)[/itex]

**The attempt at a solution**

If I determine the basis of [itex]R(A) \cap N(A)[/itex], I can determine its dimensionality and then compare it with [itex]\dim R(A) - \dim R(A^2)[/itex].

I've been unsuccessful at finding a basis. Also, given that [itex]\dim R(A) = m[/itex], is there a way to determine what [itex]\dim R(A^2)[/itex] is?