1. Suppose that

Which of these subsets of the vector space R⁴ are subspaces?

A. None

B. X only

C. Y and Z only

D. X, Y, and Z

E. the correct answer is not gieven by A, B, C, D

Solution: The answer is C

The set X cannot be a subspace because the condition b - c = 3 prevents the 0-vector from being in X.

As for Y, since a = b + c, any vector in Y can be decomposed as

Thus Y is the subspace generated by the three vectors displayed on the right side of this last equation.

Going on to Z, since a = 0 and b = d, any vector in Z can be written in the form

Thus Z is the subspace generated by the two vectors displayed on the right side of this last equation.