Suppose w is a subspace of set of real numbers r ℝn spanned by n nonzero orthogonal vectors. explain why w equals = set of real numbers r ℝn. the goal is to show that the subspace w is actually all of set of real numbers r ℝn. in general, when are two subspaces known to be the same space?

The management of a supermarket wants to adopt a new promotional policy of giving a free gift to every customer who spends more than a certain amount per visit at this supermarket. the expectation of the management is that after this promotional policy is advertised, the expenditures for all customers at this supermarket will be normally distributed with a mean of $95 and a standard deviation of $20. if the management wants to give free gifts to at most 10% of the customers, what should the amount be above which a customer would receive a free gift?