2006 - 2007 Spring MT1 - v 1 =    1 1   ...

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Last Name: Name: Student No.: Department: Section: Signature: Code: Acad.Year: Semester: Date: Time: Duration: M E T U Department of Mathematics Basic Linear Algebra Final Math 260 2006-2007 Spring 29.5.2007 16:40 120 minutes 4 QUESTIONS ON 4 PAGES TOTAL 80 POINTS 1 2 3 4 Please carefully write the logical steps leading to your answers. Correct answers without any correct reasoning will not get any points. 1. (20 pts) Let V be the vector space of polynomials in x of degree less than or equal to 3. Let ( f | g ) = Z 1 0 x 2 f ( x ) g ( x ) dx (a) Show that ( | ) is an inner product. (b) Find an orthogonal basis of the subspace S of V spanned by 1 ,x 2 and x 3 . (c) Find the orthogonal projection of x to the subspace S .
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2. (20 pts) Let T : R 4 R 4 be the linear transformation such that T ((1 , 1 , 0 , 0)) = (1 , 0 , 0 , 1), T ((1 , - 1 , 0 , 0)) = (0 , 1 , 1 , 0), T ((0 , 0 , 1 , 1)) = (1 , 1 , 1 , 1), T ((0 , 0 , 1 , - 1)) = (1 , 1 , 0 , 0). (a) Find T (( x 1 ,x 2 ,x 3 ,x 4 )). (b) Find a basis for Ker ( T ). (c) Find a basis for Range ( T ). (d) Find the matrix of T with respect to the standard bases { (1 , 0 , 0 , 0) , (0 , 1 , 0 , 0) , (0 , 0 , 1 , 0) , (0 , 0 , 0 , 1) } in the domain and range.
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3. (20 pts) A 3 × 3 matrix A has eigenvalues 1 , - 1 and 2 with corresponding eigenvectors
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Unformatted text preview: v 1 =    1 1    , v 2 =    1 1    , and v 3 =    1 1    . (a) Is A invertible? (b) Is A diagonalizable? If it is, determine an invertible matrix P and a diagonal matrix D such that D = P-1 AP . (c) Find A . (d) Find Tr ( A ) and det ( A ). 4. (20 pts) (a) Suppose that A 2 = I . Show that every eigenvalue of A is equal to 1 or-1. (b) Let V = R 3 and W be the z-axis in R 3 . Let T : V → W be orthogonal projection. Choose bases for V,W , and find the matrix of T with respect to these bases. (c) Show that for any real numbers x 1 ,x 2 ,y 1 ,y 2 , 2 x 1 y 1 + 2 x 1 y 2 + 3 x 2 y 2 ≤ q 2 x 2 1 + 2 x 1 x 2 + 3 x 2 2 q 2 y 2 1 + 2 y 1 y 2 + 3 y 2 2 (Hint: first define an appropriate inner product on R 2 )...
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This note was uploaded on 11/12/2010 for the course MATHEMATIC 260 taught by Professor Uguz during the Spring '10 term at Middle East Technical University.

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2006 - 2007 Spring MT1 - v 1 =    1 1   ...

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