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sol-2009midterm

# sol-2009midterm - 1 a A field is a set F equipped with two...

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1 1. a) A field is a set F equipped with two binary operations “addition” and “multiplication” satisfy the following properties: addition: AA1: a + ( b + c ) = ( a + b ) + c AA2: a + b = b + a AA3: 0, s.t. a + 0 = a AA4: a , s.t. a + (− a ) = 0 multiplication: SS1: a ( b c ) = ( a b ) c SS2: a b = b a SS3: 1, s.t. a 1 = a . SS4: a −1 , s.t. a a −1 = 1 for a ≠ 0 b) 2. a) A subset W of a vector space V is a subspace of V if W is a vector space, where addition and scalar multiplication of vectors in W produce the same vectors as these operations did in V. b) Nonempty: Since W 1 and W 2 are subspaces of V, 0 W 1 , 0 W 2 , then W W W 2 1 0 Closure under addition: 2 1 , W W W v u , 1 , W v u and 2 , W v u . Since W 1 and W 2 are subspaces of V, 1 W v u and 2 W v u . Therefore, 2 1 W W W v u . Closure under scalarmultiplication: W u ,and R r 1 W u and 2 W u . Since W 1 and W 2 are subspaces of V, 1 W u r and 2 W u r . Therefore, 2 1 W W W u r . So W is a subspace of V. 3. 1) Since T(1) = 0, T(x) = x, T(x 2 ) = 2 + 2x 2 = -4(x) + 2(1 + x) + 2(x + x 2 ), T(x 3 ) = 6x + 3x 3 , The matrix representation of T with respect to B is 3 0 0 0 0 2 0 0 0 2 0 0 6 4 1 0 A . 2) B x x x T 3 2 2 3 4 3 4 4 1 1 2 3 4 3 0 0 0 0 2 0 0 0 2 0 0 6 4 1 0 Therefore, T(4 + 3x + 2x 2 + x 3 ) = 1(x) + 4(1+ x) + 4(x + x 2 ) + 3x 3 = 4 + 9x + 4x 2 + 3x 3 0 1 t 1+ t 0 0 0 0 0 1 0 1 t 1+ t t 0 t 1+ t 1 1+ t 0 1+ t 1 t

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2 4. 1) F 2) T 3) F 4) F 5) T 6) T 7) T 5. 1) b 2) c 3) d 4) d 5) a 6. Yes. Let 3 3 2 2 1 1 1 , 1 , 1 t t t t t t V and R s r , . V t t t t t t t t 2 1 2 1 2 2 1 1 1 1 1 and V rt rt t t r 1 1 1 1 1 1 A1:
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sol-2009midterm - 1 a A field is a set F equipped with two...

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