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Philadelphia university Department of basic Sciences Final exam(linear algebra 250241) Student Name:……………. Date:22/1/2009 Student Number:…………. Instructor :Dr.Rahma Aldaqa Q1:(12 pts) Mark each statement as true or false 1. ( 2. ( 3. ( )2x+3y+ex =-3 is not Linear equation ) T(x,y,z)=(3,y+z,x-y+z) is linear transformation from R3 to R3 )If A and B are a square matrix of order n, then 4. ( )The system –x+y=2 has no any solution 2x-2y=-4 5. ( ) If are eigenvalues of a matrices A, then are eigenvalues of 6. ( ) If A is a matrix of order 7x9 and rank(A) is 5, then the nullity of is 4 7. ( ) 8. ( ) is a normal basis for is invertible matrix 9. ( ) (-5,4) is a Linear combination of (-1,2) and (2,3) 10.( ) Ahomogeneous system of Linear equations may be inconsistent 11.( ) The enteries of the main diagonal in a skew symmetric matrix must be all zeros 12. ( ) {(1,2,3),(0,0,0),(3,5,1)} is independent set in Q2:(6 pts) Use Gauss-Jordan Elimination to solve the following system of Linear equations: x-3y+z-w=4 2x+y+2z+w=5 -2x+6y-2z+2w=-8 Q3:( 8 pts) Let have Euclidean inner product Use GramSchmidt process to transform the basis into orthonormal basis. Q4: (4 pts) let u,v and w be vectors in inner product space such that <u,v>=3 and <v,w>=-5,<u,w>=4, Evaluate <u-v+3w,5u+v-2w> Q5( 10 pts) Let 1. Find the eigenvalues and the eigenvectors of A 2. For each eigenvalues λ Find a basis for the nullspace of (λI-A) 3. Diagonalize A and compute Q6(10 pts) Let T: T( be a Linear operator defined by 1. Find a basis for Ker(T)and a basis for Range(T) 2. Determine whether T is one-to-one, if so find 3. What is the dimension of Ker(T)and Range(T)

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