2011test1web - M346 First Midterm Exam, February 17, 2011 3...

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Unformatted text preview: M346 First Midterm Exam, February 17, 2011 3 1 1 1) (15 points) Consider the vectors 1 , 2 and 1 in R3 . Are these 5 5 3 3 vectors linearly independent? Do they span R ? Do they form a basis for R3 ? 2. (15 points) Let V =R2 [t] be the space of quadratic polynomials in a variable t and consider the linear transformation L(p) = (t + 1)p (t) from V to itself, where p (t) is the derivative of p(t). Find the matrix of this linear transformation with respect to the (standard) basis {1, t, t2 }. 1 −1 −1 1 8 1 2 8 3 7 3. Let A = . A is row-equivalent to 1 2 8 −2 −28 1 5 17 0 −29 1 0 2 0 −4 0 1 3 0 −5 . 0 0 0 1 7 0000 0 a) Find a basis for the null space of A. b) Find a basis for the column space of A. 1 5 3 . Let E be be a basis, and let x = , 4. a) In R2 , let B = −2 8 5 the standard basis. Compute the change-of-basis matrices PEB and PBE and compute the coordinates of x in the B basis. b) In R1 [t], let D = {3 + 5t, 5 + 8t} and let p(t) = 1 + t. Compute [p]D . 32 . (You do not have to 5. a) Find the characteristic polynomial of 51 compute the eigenvalues or eigenvectors). 1 −4 . (You do not have to find the eigenvecb) Find the eigenvalues of 11 tors). 311 c) λ = 2 is one of the eigenvalues of 1 3 1 . Find a basis for the 113 corresponding eigenspace. 1 ...
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This note was uploaded on 04/10/2011 for the course M 346 taught by Professor Radin during the Spring '08 term at University of Texas at Austin.

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