We can prove this by using a Theorem known as the Kernel Rank Theorem which states that the dimension of V is equal to the sum of the dimension of the range of T and the dimension of the null space of T, where T is a linear transformation.

Since the range null space of T are identical, they each have a dimension of say k, where k is a natural number. This means the sum of their dimensions is 2k which the Kernel Rank Theorem says is equal to n.