201mt2sln - B U Department of Mathematics Math 201 Matrix...

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B U Department of Mathematics Math 201 Matrix Theory Date: May 06, 2008 Full Name : Time: 17:00-18:40 Math 201 Number : Student ID : Spring 2008 Second Midterm Solutions IMPORTANT 1. Write your name, surname on top of each page. 2. The exam consists of 4 questions some of which have more than one part. 3. Read the questions carefully and write your answers neatly under the corresponding questions. 4. Show all your work. Correct answers without sufficient explanation might not get full credit. 5. Calculators are not allowed. Q1 Q2 Q3 Q4 total 10 pts 14 pts 21 pts 15 pts 60 pts 1.) Let A be a 3 × 3 invertible matrix. a) [4] Write bases for the 4 fundamental subspaces of A . (Justify your answer.) Solution: Note that since A is invertible rank ( A ) = 3 and therefore Row(A)=Column(A)= R 3 . Null space and left nullspace have dimension zero and therefore their bases are empty. Row space basis=column space basis = 1 0 0 , 0 1 0 , 0 0 1 . b) [6] Write bases for the 4 fundamental subspaces of the 3 × 6 matrix B = [ A A ]. (Justify your answer.) Solution: A is row equivalent to I. Row space basis of B = 1 0 0 1 0 0 T , 0 1 0 0 1 0 T , 0 0 1 0 0 1 T Column space basis of B = 1 0 0 , 0 1 0 , 0 0 1 To find the nullspace we need to solve Bx = 0 [ I I ] x = 0 which gives Nullspace basis of B = 1 0 0 - 1 0 0 T , 0 1 0 0 - 1 0 T , 0 0 1 0 0 - 1 T Left nullspace basis is empty since dim(left nullspace)=3-3=0.
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2.) Let P n be the vector space of polynomials of degree less than or equal to n . Let T : P 2 P 4 be the transformation T ( q ( x )) = x 2 q ( x ) for all q ( x ) P 2 . a) [4] Show that T is a linear transformation b) [4] Find the matrix of T with respect to the standard basis of P 2 .
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