The question asks whether the union of two sets, ( B ) and ( C ), is a basis for ( \mathbb{K}^n ), where ( B ) is a basis for the null space of a matrix ( A ), ( Nul(A) ), and ( C ) is a basis for the column space of the transpose of ( A ), ( Col(A^T) ). We need to determine the truth of this statement for two different fields ( \mathbb{K} ): the real numbers ( \mathbb{R} ) and the complex numbers ( \mathbb{C} ). To justify the answers, we can use the Rank-Nullity Theorem and the properties of