Math 110 - Spring 2000 - Novik - Midterm 1

# Math 110 - Spring 2000 - Novik - Midterm 1 - MON 15:28 FAX...

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Unformatted text preview: 09/25/2000 MON 15:28 FAX 6434330 MOFFITT LIBRARY 001 MATH “0 ’l Llnear Algebra) Ewing 1000 l. Novi |< Midierm I 1. (1813135) This part consists of 6 questions. Each question is worth 3pts. In each question give an example with the required properties or explain why such an example does not exist. (a) A vector space over R of dimension 100. (b) Two isomorphic vector spaces, one of which has dimension 17 and the second one has dimension 13. (c) A linear transformatiOn T : P4 (R) —) M2x3(R) which is one-to-one. (d) A generating set for P2 (R) which is not a basis. (e) An inﬁnite—dimensional vector space. (f) A linear transformation T : R80 —+ R170 of rank 90. 09/25/2000 MON 15:28 FAX 6434330 MOFFITT LIBRARY 002 2. (16pts) Suppose thah V is a. vecLo: space of dimension 9 and that W is a vector space of dimension 11. Let; T be a linear transformation T : L(V, W) a L(V, W). Show that the nullity of T cannot equal the rank of T. 09/25/2000 MON 15:28 FAX 6434330 MOFFITT LIBRARY 003 3. (1613135) Let V be :1 meta space over R. Prove that every non-zero linear transformation T : V —) R is onto, and that every non-zero linear transformation 5 : R —> V is one-to—one. 09/25/2000 MON 15:29 FAX 6434330 MOFFITT LIBRARY 004 4. (50pts) Let W1, W2 be subspaces of a vector space V. Deﬁne the sum of W1 and W21 denoted W1+W2, to be the set {\$+y 1 cc 6 W1 and y E W2}. (a) (10pts) Prove that W1 + W2 is a. subspace of V that contains both W1 and W2. (b) (10pts) Let B1 he a basis for W1, let 32 be a basis for W2, Show that Bl U 32 generates W; + W2. (0) (20pts) Suppose that Bl and .32 are disjoint. Prove that 31 U B2 is a basis for W1 + W2 if and only if W1 0 W2 = ((1) (1013135) Prove that if V is ﬁnitcedimensional, then dim(W1 + W2) 3 dim(W;) ! dimU/Vg). When does equality hold? ...
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