3. (12 points) Let Show that the system of linear equations Ax = b has a solution for all b if and only if the system of linear equations Atx = 0 has no nonzero solutions.

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 LA denotes the linear 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 linear transformation. Prove that dim R(T)+dim N(T) = dim V if V is fnite-dimensional.

3. (12 points) Let Show that the system of linear equations Ax = b has a solution for all b if and only if the system of linear equations A^tx = 0 has no nonzero solutions.

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