**Unformatted text preview: **05/12/2003 MON 10:22 FAX 6434330 MOFFITT LIBRARY 001 MATH 116— EXAM 1 PROFESSOR TARA S. HOLM September 27, 2002 Time limit: 50 minutes Name: SID: You may not consult any books or papers. You may not use a calculator or any other
computing or graphing device other than your own head! Please write your name on EVERY page of this exam. (This guarantees 4 points if you do
it, and you lose 4 points if you do not!) There are two extra problems at the end of the exam. They are not required, and you will
not receive extra credit for them. They are meant for your amusement. There are two extra blank pages at the end of the exam. You may use these for computa—
tions, but I will not read them. Please transfer all final answers to the page on which the
question is posed. 1 25 points
2 25 points 25 points 25 points 05/12/2003 MON 10:22 FAX 6434330 2 EXAM l 1. MOFFITT LIBRARY NAME: Short answer. Please answer the following questions. Please note that for the True/False questions, you are required to justify your
answer: give a proof or counterexample. If there is a computation, please circle your final answer. Let AT denote the trans-
pose of A, and tr(A) the trace of A. (a) Let V be a vector space over a field 1?, and S Q V a subset of V. What do you
need to prove in order to Show that S is a basis of V?
(b) True or False: Suppose V is a vector space. Let U and W be subspaces of V. 'l‘hen U U W is also a subspace of V.
(c) True or False: Consider the map T Z nggwl‘) —) M3X2(IF) that sends a matrix A to its transpose T(A) 2 AT. This is a linear transforma— tion.
Bases. The set 10 01 01 0?, 00 30 —-11 11 is a basis for M2x2(R) as a vector space over IR (you do not need to prove this).
Subspaces. Let ]F be a field, and consider the vector space ManGF) over F. For
every scalar a 6 13‘, define M. x {A e Mnxdr) ‘ MA) : a}. Dimension. Let W1 and W2 be subspaces of a ﬁnite-dimensional vector space V.
Recall that the sum of W1 and W2 is
W1+W2 = {n+vlu 6 W1 andvE W2}.
Prove that
dim(W1 + W2) 2 dim(i/V;) + dim(W2) ~ dimU/Vl 0 W2). 002 ...

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