Dr. A. Betten Spring 2008
MATH 369 Linear Algebra
Midterm # 2
Problem # 1
Consider the linear map L:R3→R1, L(x, y, z) = x+y+z . Compute a basis for the nullspace.
Problem # 2
Let Abe a nonsingular matrix with eigenvalue λ.Show that λ−1is an eigenvalue of A−1.
Problem # 3
Compute the eigenvalues and eigenvectors of the following matrix:
!5−1
3 1 "
Problem # 4
Compute the determinant of the following matrix:
y+z x +z x +y
x y z
1 1 1
Problem # 5
Are the following vectors dependent of independent?
a)
2
1
1
,
3
2
3
,
1
2
2
,
3)
1
−1
3
,
−5
8
−7
,
−7
13
−5
.
Problem # 6
Compute a basis for the solution space of the following system
2x1+3x2+x3+x5= 0
−3x1+x2+x4+2x5= 0
Problem # 7
Express the vector "vin terms of the basis "
b1,"
b2,"
b3of R3.
"v=
−27
1
2
,"
b1=
2
−1
3
,"
b2=
−1
3
1
,"
b3=
4
2
−1
,
