問題詳情

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 entriesin F. If T is a linear transformation, R(T) denotes the range of T, and N(T) denotesthe null space of T.

denotes the transpose of A, and LA denotes thelinear transformation from Fn to Fm that sends each vector


 1. (12 points) Let V and W be F-vector spaces, and let T: V →W be a lineartransformation. Prove that dim R(T)+dim N(T) = dim V if V is fnite-dimensional.

參考答案