MATH 219 Final, Spring 2001

AMERICAN UNIVERSITY OF BEIRUT Spring 2000-01 MATHEMATICS 219 Time : 2 Hours. FINAL EXAMINATION Prof. H. Abu-Khuzam JUNE 20 ,2001 1. Let T: P2(R) > P3(R) be the linear transformation defined by T( p(x) = x7 pW) + 3p) a. Find the matrix of T relative to the standard ordered bases of P2(R) and P3(R). [ 6 points] b. Find a basis for Ker T. Find dim (Im T). { 6 points] 5.1.0 0 0 6 a. Find the eigenvalucs and a basis for cach eigenspace of A. [ 8 points] b. Deduce that there is a basis of R® consisting of eigenvectors of A and find an invertible matrix P and a diagonal matrix D such that P'AP=D. ( Do not verify) [ 4 points] ca 2c. Use part (b) above to find an orthonormal basis of R* consisting of eigenvectors of A. (3 points] 3. Let A and B be nxn matrices such that AB is invertible. Prove that A must be invertible. ( Hint: Use Determinants) [7 points] 4. Show that an orthogonal set of nonzero vectors in an inner product space V is linearly independent. { 8 points] 5. Let T: VV be a linear operator on a finite dimensional vector space V. Show that (Ker T) 9 (Im T) ={0} = V= (Ker T)+ (Im T) [7 points] h 6. Use the row echelon form of the augmented matrix of the following system x+2y+z=3 xX+3y- 224 x + 2y4+(a-8)z=a. to find the value(s) of a for which the system has a. no solution b a unique solution c. infinitely many solutions. [ 10 points] 5 7. Let T: VOW be a linear transformation. Show that If Ay, Ao, ..., A, are vectors in V, such that T(A;), T(A2), ..., T(An) are linearly independent in W, then Aj, A2,..., A, are linearly independent in V. [ 8 points] 8. Let V be a vector space of dimension 3, and let U be a subspace of V of dimension 2. Show that there exists a lincar transformation T; VV such that T(A)=0 for all AeU, and T is not the zero map on V. [6 points]. Answer TRUE or FALSE only: [ 3 points for each correct answer , 0 for no answer, and ~1 for each wrong answer]. Let A be a 4x4 matrix such that (rank A)=3, then |A] = 0 The subspace of R spanned by the vectors (1,0 ,-1). (2.1, -3). and (5,2,-7) has dimension 3. Similar matrices have the same deteminant. Any set of 2 vectors in R* can be extended to a basis of R? . If N is a square matrix such that N°=0), then I-N is invertible The space of all symmetric 3x3 matrices has dimension 5. If A, B, and C are nxn matrices such that AB=AC. then B=C. Let W be a subspace of a vector space V, then the set U= {Ae V: A¢W} is a subspace V. Let A be an nxn matrix such that AA'=I , where [ is the identity nxn matrix , then JA] = 1. [ 27 points]