mid-ex_sol - SOLUTIONS FOR MIDTERM 1 (10 points) : Find the...

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Unformatted text preview: SOLUTIONS FOR MIDTERM 1 (10 points) : Find the equation of the plane, containing the points (2 , 1 , 1), (1 , 1 ,- 1), and (0 , 3 , 3). The set of all points of the form (2 , 1 , 1) + t ( (2 , 1 , 1)- (1 , 1 ,- 1) ) + s ( (2 , 1 , 1)- (0 , 3 , 3) ) = (2 , 1 , 1) + t (1 , , 2) + s (2 ,- 2 ,- 2) = (2 + t + 2 s, 1- 2 s, 1 + 2 t- 2 s ) , with t, s ∈ R . 2 (10 points) : Show that { f ∈ F ([0 , 1] , R ) : f (0) = f (1) } is a subspace of F ([0 , 1] , R ). We have to show that V = { f ∈ F ([0 , 1] , R ) : f (0) = f (1) } is closed under addition and multiplication. That is, if f, g ∈ V , and c ∈ R , then cf and f + g also belong to V . To handle cf , note that [ cf ](0) = cf (0) = cf (1) = [ cf ](1), hence cf ∈ V . Similarly, [ f + g ](0) = f (0)+ g (0) = f (1)+ g (1) = [ f + g ](1), which implies f + g ∈ V . 3 (10 points) : Determine whether the polynomial x 3 + x + 2 can be expressed as a linear combination of x 3 + x 2- x + 1 and 2 x 3 + 3 x 2- 4 x + 1....
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This note was uploaded on 02/02/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

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mid-ex_sol - SOLUTIONS FOR MIDTERM 1 (10 points) : Find the...

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