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# 2009midterm - Term Test MATB24 Linear Algebra II 2009 1(10...

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Term Test MATB24 Linear Algebra II 2009 1. (10 points) . a) Give the definition of a field. b) Let K be a vector space over Z 2 with basis {1, t}, so K = {a + bt | a, b Z 2 }. It is known that K becomes a field of four elements if we define t 2 = 1 + t. Write down the multiplication table of K. 2. (12 points) a) Give the definition of a subspace of a real vector space V. b) Let W 1 and W 2 be subspaces of a vector space V. Let 2 1 2 1 , | W W W w W w w W be the intersection of W 1 and W 2 . Show that W is also a subspace of V. 3. (12 points) Let T: P 3 → P 3 be the linear transformation defined by T(p(x)) = D 2 (p(x)) + xD(p(x)) (D 2 -- the second derivative) 1) Find the matrix representation of T relative to B = {1, x, x 2 , x 3 } and B = {x, 1 + x, x + x 2 , x 3 }. 2) Use the result in part 1) to compute T(4 + 3x + 2x 2 + x 3 ). 4. (14 points) True or False: 1) If T is a linear transformation from V to V, then the intersection of range(T) and Ker(T) must be   0 .

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