test2-soln-M340L-Fall2007

# test2-soln-M340L-Fall2007 - TEST 2 M 340L (Fall 2007,...

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M 340L (Fall 2007, 60150) T EST 2 Friday, November 9, 2007 1 Name: UT EID: INSTRUCTIONS 1. Please LEGIBLY print your name and UT EID in the spaces provided above. 2. The total number of points on this Test is 100 points. It will be graded out of 100 points. 3. Show all work for full credit. Unjustiﬁed answers may yield no points. 4. There are 11 questions and 10 printed pages in this Test. 5. Calculators with no communications capabilities are permitted. All other forms of aid, including textbooks and notes, are not allowed. Question Score Subtotal Question Score Subtotal 1 5 6 10 2 5 7 10 3 5 8 10 4 5 9 10 5 10 10 10 11 10 Total 100

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M 340L (Fall 2007, 60150) T EST 2 Friday, November 9, 2007 2 1. a) (5 pts) Exhibit a subset of R 2 which is closed under scalar multiplication but not addition. The union of the horizontal and vertical axes. b) (5 pts) Exhibit a subset of R 2 which is closed under addition but not scalar multiplication. ± ² x 0 ³ R 2 ´ ´ ´ ´ x Z µ 2. (5 pts) Suppose f : R 3 -→ R is linear and f is not the zero map. Prove that f is surjective. Since codomain( f ) = R , dim image( f ) can only be either 0 and 1 . Since f is not the zero map, dim image( f ) > 0 . Thus dim image( f ) must be 1 , which is the dimension of R . So, image( f ) = R , i.e. f is surjective.
M 340L (Fall 2007, 60150) T EST 2 Friday, November 9, 2007 3 3. a) (5 pts) Is the unit circle in R 2 centered at the origin a subspace of

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## This note was uploaded on 03/22/2008 for the course M 340L taught by Professor Pavlovic during the Fall '08 term at University of Texas at Austin.

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test2-soln-M340L-Fall2007 - TEST 2 M 340L (Fall 2007,...

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