Two theorems relevent to the problems:
These are but two of many interesting properties which are obscured when one simply ignores the dimensional structure of vectors and matrices and works with numbers alone. A simple example solving both pop-quiz problems is
For proofs of the two theorems above, and a full discussion of the differences between traditional linear algebra and the algebra which scientists and engineers really use, see my book:
George W. Hart, Multidimensional Analysis: Algebras and Systems for Science and Engineering, Springer Verlag, 1995.