Final Exam for Math 310: Applied Linear Algebra (22632)
December 12, 2007
Problem 1. (20pts) Let L:R3→R3be the linear transformation
L
x1
x2
x3
=
4x1+x2+ 2x3
0
−3x1−x3
.
Find a matrix A∈R3×3such that L(x) = Ax. Determine the fixed points of L, i.e. apply
Gauss-Jordan reduction to the linear system Ax =xand determine a basis for its solutions.
Problem 2. (20pts) Let
A=
2−1 0
−120
0 0 1
.
Determine the eigenvalues and eigenspaces of A. Find an orthogonal matrix U∈R3×3such
that UTAU is diagonal. What is the general solution to the system of linear differential
equations y0=Ay?
Problem 3. (20pts) Let
A=
−1−2
−1−1
1−1
1 0
and b=
−1
1
0
0
.
Apply the Gram-Schmidt algorithm to compute the QR-factorization of A. What is the least
square solution to Ax =b? Find the projection and distance of bto R(A).
Problem 4. (20pts) Find the dimensions and bases for the row space, column space and
null space of
A=
1−3−2
2−1 1
−2 0 −2
.
Problem 5. (20pts) Let S= span(v1, v2, v3), where
v1= [1,1,−1]T, v2= [−2,2,−3]T,and v3= [2,2,−2]T.
Determine the dimension and a basis for Sand S⊥. What is the geometric description of S
and S⊥?
