2011sol1 - M346 First Midterm Exam Solutions, February 17,...

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Unformatted text preview: M346 First Midterm Exam Solutions, February 17, 2011 1) (15 points) Consider the vectors 1 1 3 , 1 2 5 and 3 1 5 in R 3 . Are these vectors linearly independent? Do they span R 3 ? Do they form a basis for R 3 ? Answer : Row-reducing the matrix A = 1 1 3 1 2 1 3 5 5 , whose columns are the vectors in question, yields 1 5 1- 2 . There are only two pivots, so the vectors are not linearly independent (since there are more than two columns), they do not span (since there are more than two rows), and they do not form a basis. 2. (15 points) Let V = R 2 [ 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, t 2 } ....
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2011sol1 - M346 First Midterm Exam Solutions, February 17,...

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