MATH 251 Midterm: MATH 304 TAMU Spring 04 Exam 2 Solutions
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March 28, 2004: determine if the following sets of vectors are or are not vector spaces. If they are not, explain why. (a) v = solution set of the equations x + y z w = 0 and x + y + 2w = 0 in r4. Solution sets of homogeneous systems of linear equations are always vector spaces. (b) w = (cid:8)[ x y ] (cid:12)(cid:12) y = x + 1 x(cid:9). So w is not closed under scalar multiplication. 2 ] w but that 2 [ 1. 4 ] / w . (c) x = set of upper triangular 3 3 matrices. We know that the set of 3 3 matrices forms a vector space. Explain how you know a set is or is not linearly independent. If a set is linearly dependent, then nd a linear dependency among the vectors. For two vectors to be linearly (a) dependent, one must be a scalar multiple of the other.
