Download Philadelphia university Department of basic Sciences Final exam(linear algebra 250241)

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Transcript
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)