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601-hw-iv

# 601-hw-iv - MATH 601 AUTUMN 2008 Additional homework IV...

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Unformatted text preview: MATH 601, AUTUMN 2008 Additional homework IV, October 18 PROBLEMS 1. For any integer n ≥ 0 , let V be the vector space of all polynomials f of the real variable x with deg f ≤ n . Determine if the linear operator T : V → V , given by ( Tf )( x ) = f ( x ) + f ( x- 1) , is surjective. 2. Let V denote the real vector space of polynomials ϕ ( t ) in the real variable t whose degree does not exceed 5, and let W be the subspace of V formed by all such polynomials that in addition satisfy the conditions ϕ (1) = ϕ (1) , Z 1 ϕ ( x ) dx = ϕ (- 1) . Construct a vector space Z and a linear operator T : V → Z for which Ker T = W . Find a basis of W . 3. Determine rank M for the matrix M = - 1 1 3 5 2 2 4 1 3 1 3 . 4. Find a basis of the subspace of R 4 consisting of all solutions [ x y z t ] T to the homogeneous system y + 3 z + 2 t = 0 , x- z = 0 , x + 2 y + 5 z + 2 t = 0 . 5. Let V be the vector space of all real polynomials ϕ ( t ) = a + bt + ct 2 + dt 3 of degree not exceeding 3 , and let T : V → R 4 be the linear operator given by Tϕ = ϕ (17) ϕ (- 781) ϕ (2533) ϕ (- 6) . Does there exist a vector y in R 4 for which the linear equation Tϕ = y is inconsistent (i.e., has no solution)? Justify your answer. 6. Let V be the vector space of all polynomials ϕ ( t ) = a + a 1 t + ... + a 7 t 7 of degree not exceeding 7 , and let T : V → R 2 be the linear operator given by Tϕ = ϕ (11)- 4 ϕ (37) ϕ (29) + 5 ϕ (61) , where...
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601-hw-iv - MATH 601 AUTUMN 2008 Additional homework IV...

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