Exam1_Math121

Exam1_Math121 - Midterm 1 for Math 121 Fall 2006 Wednesday...

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Midterm 1 for Math 121, Fall 2006. Wednesday October 25 Time allowed: 53 minutes Apart from the first problem you must fully justify your answers. You may assume all vector spaces are finite-dimensional unless otherwise stated. Recall that for a linear operator T : V V , the notation N ( T ) denotes the null space and R ( T ) denotes the range. 1. (20 points) Mark the following statements true or false. No justification is needed. (a) Let W 1 ,W 2 be two vector subspaces of V and β 1 2 be bases of W 1 ,W 2 respec- tively. Then β 1 β 2 is a basis of W 1 W 2 . (b) Let P ( R ) denote the (infinite-dimensional) vector space consisting of all poly- nomials with real coefficients. Let g ( x ) ∈ P ( R ) be some fixed polynomial. Then the function T : P ( R ) → P ( R ) given by T ( p ( x )) = g ( x ) p ( x ) is linear. (c) Let V be a vector space and T,U : V V be two linear operators. Then N ( T ) N ( TU ). (d) Let
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This note was uploaded on 09/09/2009 for the course MATH 350 taught by Professor Hoelscher during the Spring '08 term at Rutgers.

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Exam1_Math121 - Midterm 1 for Math 121 Fall 2006 Wednesday...

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