申論題

Notation: R denotes the feld of real numbers; C denotes the field of complex numbers. F denotes an arbitrary field; denotes the set of all m x n matrices with entries in F. If T is a linear transformation, R(T) denotes the range of T, and N(T) denotes the null space of T. denotes the transpose of A, and L_A denotes the linear transformation from Fⁿ to F^m that sends each vector
 1. (12 points) Let V and W be F-vector spaces, and let T: V →W be a linear transformation. Prove that dim R(T)+dim N(T) = dim V if V is fnite-dimensional.