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MATH311 practice2

# MATH311 practice2 - or false circle your answer You do not...

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Math 311 second practice exam Instructions: This is a closed-book exam. You have 50 minutes. Use the backs of the pages for scratch paper; if you attach more pages, write your name on every extra page you use. Make sure to show all your work, and provide clear details. Each problem is worth 10 points. 1. Let D be the differentiation operator, and I the identity operator. Let u be the function u ( x ) = cos 2 x . Find ( D 2 - I ) u. 2. Show that the polynomials { 1 , x + x 2 , x 2 } form a basis for the vector space P 2 of polynomials of degree at most 2 . 3. Find a basis for the following subspace of R 3 : Span 1 2 3 , 1 4 5 , - 1 2 1 . What is the dimension of this subspace? 4. Find the eigenvalues and the corresponding eigenvectors for the matrix 2 - 2 1 0 1 3 0 0 1 . 1

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5. Indicate whether the following statement is true
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Unformatted text preview: or false ; circle your answer. You do not need to explain or justify your answer. (a) (2 points) The set of functions f in C [0 , 1] such that f (0) = f (1) is a subspace of C [0 , 1] . T F (b) (2 points) The linear transformation f : R 3 → R 2 given by f x y z = ± 1 2 3 4 5 6 ² x y z is one-to-one. T F (c) (2 points) A linear transformation from R 4 to R 4 can have an image of dimension 2 and an null space of dimension 1 . T F (d) (2 points) There is a basis for R 4 consisting of eigenvectors of the matrix 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3 . T F (e) (2 points) The vectors 1-1 1 , 1 2 1 span R 3 . T F...
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