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midterm1

# midterm1 - Harvard University Midterm 1 for Math 121 Fall...

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Harvard University Midterm 1 for Math 121, Fall 2007 Monday, October 22, 2007 Time allowed: 53 minutes 3 pages (including this one) Last Name: First Name: Harvard ID Number: Apart from question 1, you must fully justify your answers. On this exam the full score is 100 points. You may assume all vector spaces are finite-dimensional unless otherwise stated. Recall that for a linear transformation T : V W , the notation N ( T ) denotes the null space and R ( T ) denotes the range. Good Luck! 1

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Harvard University Midterm 1 for Math 121, Fall 2007 Monday, October 22, 2007 1. (20 points) Mark the following statements true or false. No justification is needed. a) Let W = { f ( x ) P ( R ) | f ( x ) = f ( - x ) } , where P ( R ) is the set of all polynomials with real coefficients. ( P ( R ) has infinite dimension.) Then W together with the common addition and scalar multiplication is a vector space over R . b) The vectors (1 , 0 , 0) , (0 , 2 , 2) , (2 , 4 , 4) in R 3 are linearly independent. c) Let V be a vector space. Let A 1 , A 2 be two subspaces of V , and let γ 1 , γ 2 be bases of A 1 , A 2 , respectively. Then γ
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midterm1 - Harvard University Midterm 1 for Math 121 Fall...

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